Is the Holy Spirit more effective at restoring people's mental health than secular therapeutic approaches and treatments?

I found this review "Religion, Spirituality, and Schizophrenia: A Review" which shows links (both good and bad) between religiousness / religious practice and outlooks for schizophrenic patients. Nothing supernatural to see here, but I have a hard time figuring out how a study for what you are actually interested in would have to look. There are a lot of papers linked in that review, maybe one of those can help answer your question, although I doubt it's properly answerable.

I want to point out what was included as "positive religious coping" and "negative religious coping", which may give a hint how, as a religious community, these people can be helped better.

Religious coping is multidimensional and refers to functionally oriented expressions of religion in times of stress. Religious coping is operationally defined as “the use of religious beliefs or behaviors to facilitate problem-solving to prevent or alleviate the negative emotional consequences of stressful life circumstances.”[52] The concept of religious coping has been refined and categorized as helpful or positive, harmful or negative, and with mixed implications. The positive religious coping strategies include religious purification/forgiveness, religious direction/conversion, religious helping, seeking support from clergy/members, collaborative religious coping, religious focus, active religious surrender, benevolent religious reappraisal, spiritual connection, and marking religious boundaries. The negative religious coping strategies include spiritual discontent, demonic reappraisal, passive religious deferral, interpersonal religious discontent, reappraisal of God's powers, punishing God reappraisal, and pleading for direct intercession.[53] The religious coping strategies with mixed implications include religious rituals in response to crisis, self-directing, deferring, and pleading religious coping.

Regarding outlooks:

Researchers have shown that religion/religiousness in patients with schizophrenia is associated with increased social integration, reduced risk of suicide attempts,[38,41] reduce risk of substance use,[38,42] decreased rate of smoking,[43] better quality of life,[10,44,45] lower level of functioning,[26] and better prognoses.[46] With regard to the relationship of religion and psychosocial adaptation, the findings are contradictory, with some reporting better psychosocial adaptation[47] and others reporting poor social and psychological status in a majority of patients.[9] Religious support and spirituality has also been found to be associated with better recovery[42,48,49] and reduced relapse rate.[47,50] However, in some patients, higher religiosity has been linked to higher risk of suicide attempt.[38]

You may want to look into this study on the effectiveness of the ministry Be In Health's flagship class at curing depression. They take the hard cases of Christian healing, and over the years The Holy Spirit has taught them a lot of things about a lot of deep and otherwise incurable diseases, disorders, syndromes, etc.

Effect of a Faith-Based Education Program on Self-Assessed Physical, Mental and Spiritual (Religious) Health Parameters

The study found that a year after participants took their class, 90% of those with depression were fully free of it. Also, although I don't have a study for it, my mom and took their class, and it has been a significant step in her journey toward recovering from chronic fatigue and my journey toward recovering from Autism.

Although it may "resolve" on it's own, many secular mental health professionals consider Multiple Personality Disorder, now known as Dissociative Identity Disorder, to be incurable. There is speculation in the literature that at least some instances of demonic possession, whether in Scripture or currently in environments amenable to such a diagnosis, are actually cases of D.I.D.

This link provides a free download in PDF form of a book called "The Shining Man With Hurt Hands.". In it the author, Ellis H. Skolfield, describes his experience and success/progress in helping individuals who have multiple personality disorder to approach and even occasionally achieve integration.

While he finds, along with secular medicine, that the vast majority of D.I.D. cases are the product of extreme past trauma (most often abusive or sexual in nature), his diagnosis is that of the incursion of spiritual entities (both good and bad) into a personality as it is fractured. He does not consider all such incursions to qualify as demonic possession.

He finds that many multiples discover a "person" inside them that they describe as a shining man with hurt hands, hence the book title. His therapeutic techniques are entirely spiritual and centered upon the name of Christ.

Mental health means ‘a mind that is healthy’. From this standpoint, a Christian would easily agree that the ideal mind was that of the god-man Jesus Christ.

However, in the world, mental health usually means a mental imbalance possibly caused by internal chemistry such as hormones, etc., or a complete inability to function in a professional career, or a criminal. In other words, mental health in the world consists in physical imbalances or handicaps as well as sinful dispositions that lead to extreme behavior such as murder, etc. On the other hand, a self-righteous, seeming balanced worldling that does not live in any criminal lifestyle, and does not have any physical handicaps but has a ‘healthy’ view of himself, might under a Christian perspective, be the most wicked of all types – so doctrinally the least healthy minded of all. Such a wanton sinner is only appearing balanced because all their lusts are getting equal attention in their lifestyle, whereas a criminal might chase only after a single lust.

As we try to look at those two groups, those with physical handicaps and those with sinful dispositions, the Bible has some obvious things to say. First, the world is in what could be described as in a state of an unhealthy tormenting stress response through fear of death. The whole world has a mental problem on this account because the Devil is a tyrant that tortures a sinners soul through fear:

14 Since the children have flesh and blood, he too shared in their humanity so that by his death he might break the power of him who holds the power of death—that is, the devil—15 and free those who all their lives were held in slavery by their fear of death. (Heb 2:14–15) NIV

Second, one cannot underestimate the healthy influence on the mind freed from that fear and a mind reconciled to our creator, thereby obtaining peace, a clear conscience and a strong sense of the deep love God has for us:

Therefore, since we have been justified through faith, we have peace with God through our Lord Jesus Christ, 2 through whom we have gained access by faith into this grace in which we now stand. And we boast in the hope of the glory of God. 3 Not only so, but we also glory in our sufferings, because we know that suffering produces perseverance; 4 perseverance, character; and character, hope. 5 And hope does not put us to shame, because God’s love has been poured out into our hearts through the Holy Spirit, who has been given to us. (Ro 5:1–5). NIV

Even a secular psychologist can understand what healthy effects ‘feeling loved’ and ‘feeling forgiven’ has on the mind and being ‘at peace’. Is this not the main element of mental health? Are criminals not often abused as children – not knowing a stable loving environment? Not only these healthy foundations but all the fruits of the Spirit strike meaning into exactly what mental health is:

22 But the fruit of the Spirit is love, joy, peace, forbearance, kindness, goodness, faithfulness, 23 gentleness and self-control. Against such things there is no law. (Ga 5:22–23). NIV

Furthermore Jesus himself taught the beatitudes and lived them by example, saying ‘Blessed is he who…’ where blessed literally mean ‘happy are those’. Now how can one experience happiness without series benefits to mental health?

In summary, to answer the question, some problems do have physical elements and a spiritual answer may not be available. For example I never heard of someone with Down Syndrome being healed and that must introduce some 'mental challenges'. Another example is that some people with high levels of anxiety can have their electron pulses from their nervous system, actually measured with instruments, as the pulse creates higher than normal signals to the parts of the brain that look for dissimilarities and create higher than normal risk assessments. Medication might help in situations like that, and good physicians are also gifts from God. However, broadly speaking of course justification by faith, the resulting new birth, the practice of meditating on God’s word, and the revelation of His love for us through the comforting Spirit, must and does produce greater health benefits for the mind than any mere mortal or human effort can ever attempt to provide. If this were not true God does not save sinners.