Was there a time when it was easier for slaves than citizens to access state-sponsored medical facilities in Ancient Rome?

The information given in the WP article seems a bit misleading and has the occasional error. The army and navy had its special valetudinaria. Therefore a soldier would have been treated there and as long slaves were in or better with the army then they would have been treated there also.

But what this entails in republican times is quite different from our modern understanding of 'military medicine':

The Roman military provided the only organized military medical services in the classical age […] Romans seem to have had little expertise and less interest in medicine. For the most part, medical science was con ned to herbalism, people were expected to be their own doctors, and what professional physicians there were generally found themselves distrusted and denigrated. The rst Greco-Roman physician whose name we know was Archagathus, who emigrated to the city in 219 B.C. and established his taberna (clinic) with state help. He showed such a proclivity for the knife and cautery that he acquired the nickname carnifex (butcher) and a general opprobrium.
That attitude changed after the fall of Corinth in 146 B.C. and the absorption of Greece into the Roman state. The subsequent flood of Greek physicians to Rome led Pliny the Elder to complain that the city had done perfectly well for 600 years without doctors and he failed to see the need for them now. The standing of Greco-Roman physicians improved dramatically in 91 B.C. when Asclepiades of Prusa came to the city. His use of diet, baths, exercise, and massage and his avoidance of surgery and poison- ous medicines enhanced both his personal reputation and that of his profession.
The republican Roman army had no formal medical corps, and wounded soldiers were bandaged by their comrades and cared for in the homes of local citizens. Soldiers on campaign were at high risk of epidemic disease. Livy (Titus Livius) described an epidemic during the siege of Syracuse in which so many soldiers died that the overwhelmed survivors stopped burying the dead and left them to rot where they fell. Legionnaires became so despondent that they ung themselves unprotected on the enemy lines, preferring to die by the sword rather than from disease. Disease was not the only problem. Livy also noted that a Ro- man soldier was more likely to die from his wounds after a battle than be killed directly during the fray. (McCallum p 270–272)

But that changed starting with Caesar's campaign in Gaul and was quite professionalised by the time of Trajan.

The second paragraph quoted in the question is quite misleading, as they at least mixed up AD and BC abbreviations, correctly noted in the question. Apollo the healer and Aesculapius were well introduced within Rome by then. And they were imported officially, by the state itself.

More serious is that the article seems to amalgamate anachronistically the non-existant duality of physical and spiritual healing, which were often found together on one and the same continuum, despite what progressivist apologetes of Hippokrates – "Father of Modern (non-spiritual) Medicine" – would like to believe. Providing a public facility for spiritual healing was one form of "medical facility". (Cf. Hortsmannshoff: "'Did the god learn medicine?' Asclepius and Temple Medicine in Aelius Aristides' Sacred Tales" p 325-342.)

The earliest references to any Roman involvement with the Greek world of health and healing are not concerned with secular healing, but with the importation of new gods to defend the Roman state during an epidemic. We have already encountered the introduction of the cult of Apollo the Healer in 433 BC, but it was the arrival of Asclepius that attracted most attention from later writers. In 293BC, after three consecutive years of plague in Rome, it was revealed, after a priestly consultation of the Sibylline Books, that the epidemic could be halted only by summoning Asclepius from the shrine of Epidaurus. The next year a formal embassy, led by Quintus Ogulnius, was dispatched there by the Roman senate; the god himself consented; and, in the form of a snake, he was conveyed to Italy. […] Several features of this account deserve notice. First, the introduction of Asclepius and his cult to Rome is a formal act by the Roman state, not an ostensibly private initiative, […] (Nutton, p 162.)

Before that, folk and family medicine and remedies, and mountain men, Marsi and snake handlers (considered experts of healing even by the later Galen!) were giving out both herbs and tincture, venoms and poisons, performing operations, prayers, chants and rituals. For sick to go or be brought for incubation near a temple of a benevolent god or one known to possess and share his healing powers was entirely common.

What our literary sources do not tell us, but has been revealed by archaeology, is that similar healing sanctuaries were common in central Italy, often near springs. At many of them, sufferers dedicated votive offerings in the form of terracotta models of the affected part of the body – feet, hands, eyes and sexual organs in particular. Incubation may even have been practised at a shrine at Lavinium, not far from Rome. Although the form the cult of Asclepius took in Rome is assuredly Greek, the arrival of Asclepius can also be interpreted as an index of the assimilation of Roman Italy as a whole into the Greek world. (Nutton, p 164.)

Rather, Hemina is emphasising a distinction between Romans and Greeks: whatever healing skills the Romans themselves might have possessed, they did not count as medicine, since that was a purely Greek import. (Nutton p 164.)

But whatever your status: having access to a real (Greek) doctor was a matter of prestige, whatever the status of the doctor. It seems to be quite well known that a human doctor was superior to incubation, and therefore preferred. In case a 'professional' wasn't available Roman family structure already provided an alternative in the pater familias being responsible for curing maladies. But going to such a sanctuary does not seem to have been restricted to any class.

‘Keep away from doctors’, Cato’s advice to his son, is not a condemnation of all forms of medicine or of all those with healing skills, but only of the Greeks or those who follow them. He approves of the Roman way of healing, by which the head of a household took responsibility for the health of all its members, animals included. He himself is said to have kept a notebook in which he wrote down prescriptions and diets for his household. This medicine is confined to the household: it is not bruited abroad, or traded for cash; and, so Cato claimed, it is effective, for it kept him and his family hale and hearty for many years. (Nutton p 165.)

Precedents from Hellenistic Greek cities almost certainly lie behind the action of Julius Caesar, perhaps around 49 BC, in conferring Roman citizenship on any doctor practising in the city of Rome, and, scarcely a decade later, behind the granting to all doctors and teachers throughout the Empire of immunity from conscription and from having troops billeted upon them. In 23BC or shortly after, following his dramatic cure by an ex-slave doctor, Antonius Musa, the emperor Augustus is said to have granted tax immunity for ever to all practitioners of medicine. The number of immigrant doctors who took advantage of this generosity to become full citizens and the procedures whereby one proved one’s qualifications as a doctor or a teacher are both unknown. (Nutton p 167.)

By contrast with the Greek East, where there were medical dynasties and, even occasionally, doctors drawn from wealthy families, the practitioners of medicine in Rome were at best parvenus, and, far more often, drawn from what in law were the very lowest sections of society, slaves and ex-slaves. Although the figures are a little distorted by the abundance of information provided by inscriptions coming from the emperor’s Roman household of slaves and ex-slaves, scarcely 10 per cent of doctors recorded epi- graphically from Italy and the western Latin provinces of the Empire before AD 100 are Roman citizens; over 75 per cent are either slaves or ex-slaves; and fewer than 5 per cent bear a non-Greek name. Although the percentage of citizens and of non-Greek names rises over the next two centuries, the general pattern remains the same. (Nutton p168.)

There are both social and intellectual reasons for this reluctance. The true practice of medicine is incompatible with the indecent pursuit of ‘quaestus’, monetary gain: for all his knowledge, Celsus and many of his intended audience were and remained gentlemen, interested at least as much in warfare and agriculture as in medicine. Like Cato, Celsus implies that one should confine one’s medical attentions to one’s family and friends, and certainly not attempt to treat large numbers of patients all over town, for one would not be able then to give them the individual attention so necessary for a cure. (Nutton p169.)

What is today understood as a 'hospital', not a little private clinic, seems to be a much later invention. Though still in 'Roman times' much of the above is true as early as the early republic, hospitals seem to belong into late antiquity.

With the legalisation of Christianity in the fourth century, the Christian duty of care for the sick and needy became ever more visible, expressed in bricks and mortar in a new architectural form – the ‘hospital’. (Nutton p 314.)

What a travelling doctor could do was inevitably limited by the drugs and instruments he had at his disposal, and by the time he could spare before moving on. ‘Whoever can cure the sick while on the move?’ wondered Seneca, disapprovingly, for ‘travel doesn’t make a man a doctor’. By contrast, a doctor resident in one town could turn his house into a surgery or a cottage hospital, where the sick might spend several days under the watchful eye of the doctor and his assistants. (Nutton p 270.)

The idea of a 'hospital' for the sick did not develop until the Byzantine period, pace H. Avalos, Illness and Health Care in the Ancient Near East (Atlanta, 1995), 184, suggesting that the Gula temple might have served as a healing centre, similar to the Asklepieion. (Horstmanshoff p 39).

Returning to the concept of valetudinarium:

Virtually all Roman medical advances came from the military rather than from civilian physicians, and the best of Roman physicians, including Dioscorides and Galen, served with the army. Roman medical educators advised those who would be surgeons to find themselves an army and follow it in battle because they would never see enough wounds in the civilian practice to become competent in their craft.
In sum, Rome brought the organization of military medicine to a level never before achieved. Its military hospitals surpassed anything again seen until the late 19th century, and the technique of its surgeons was probably as good as it could have been with neither anesthesia nor antisepsis. (McCallum p 273.)

Conclusion

Lacking complete surveys: it might be argued that indeed there was a time when there was one type of state sponsored health service unavailable to ordinary citizens within the walls of Rome. But there are certain limits to this conclusion:
This only holds true for the hands-on military medicine in the imperial time, available only to those who where with the army. – All other types of medicine and care, private or public, seem to have been available to all classes most of the time, in principle.
'The state' being a problematic concept in itself, when applied to the Roman Republic or Empire, did provide direct and indirect sponsoring: temples, baths and markets (where healers practised), exempted doctors from duties like taxes and service, regardless where they served the public good (that is: health).
Regarding state sponsoring: for example the public buildings of 'the state', the res publica, were largely funded by private benefactors, not the tax payers as modern readers might expect. It would therefore be quite peculiar to demand that rich benefactors paying into a public pool would count as contributing to "state-sponsored medical facilities" but patrons covering the costs for their family members and slaves would not.
As shown above having access to a type of 'hospital' (in the modern sense implied in the Wikipedia article) does not say anything about the quality of care to expect there. Since the early forms were indeed with the military, and the military practitioners had horrendous track records before Caesar, being sent there was no advantage at all. The same holds true when the Empire began its steady decline in the 3rd century when it was again preferable to be treated at home or at least away from the camps and garrisons.
Since this was specifically mentioned in the question: "the poor had to resort to spiritual aid". Not only was spiritual healing a perfectly accepted fine form of healing. This "last resort" holds also true for the rich in many cases when the doctors just couldn't do much more then to comfort the sick and maybe alleviate some symptoms. Much of the tradition of family and folk medicine from Roman times is lost now. In extant writings acknowledging their wisdom and successes we might conclude that the gaps between just waiting, using folk medicine and seeking a Greek professional might have been much smaller than we like to expect.

David Wootton argues, from the fifth century BC until the 1930s, doctors actually did more harm than good:.
Although we believe that people who choose to live high-pressure lives may bring on heart attacks, we rarely blame people for getting cancer or arthritis; ancient medicine, by contrast, implied that all diseases reflected deficiencies in lifestyle. In some respects this empowered patients. Are you an old man who wants to make love to a young woman? Then eat the right food first–– pigeon breasts are particularly recommended. (p 56.) [Or to be even more polemical] Put simply, the fundamental puzzle about medicine from the fifth century bc until the end of the nineteenth century is that doctors found patients who were prepared to pay for treatment that was at best ineffectual, and usually deleterious. (p 67.)

_{Although this is getting a bit lengthy already and prone to misunderstandings, the following from the same page has to be quoted as well: "But the fact that there was no progress––far too little to have any systematic impact on life expectancy––and the fact that medical intervention did more harm than good, does not mean that doctors did not cure patients. "}

Quotes from:

H. F. J. Horstmanshoff & Marten Stol: "Magic and Rationality in Ancient Near Eastern and Graeco-Roman Medicine", Studies in Ancient Medicine 27, Brill: Leiden, Boston, 2004.

Jack E. McCallum: "Military Medicine", ABC-Clio: Santa Barbara, Denver, 2008.

Vivian Nutton: "Ancient Medicine", Routledge: London, New York, ²2013.

David Wootton: "Bad Medicine: Doctors Doing Harm Since Hippocrates", Oxford University Press: Oxford, New York, 2006.