What was the point of the third article of Remonstrance?

When I first read your question my immediate thought was it is not what is said here that shows the conflict with Calvinism but what is 'not said'. Interestingly enough I was trying to find proof of this impression and found the very same observation made my a modern Remonstrant with respect to the seventh article of the Methodist church.

Original sin standeth not in the following of Adam (as the Pelagians do vainly talk), but it is the corruption of the nature of every man, that naturally is engendered of the offspring of Adam, whereby man is very far gone from original righteousness, and of his own nature inclined to evil, and that continually. (VII Article, Methodist Church)

What may not first be obvious on the Methodist article is that the 'inclination to evil' is not an actual sinful depravity and personal wickedness at birth but more like a severe weakness that will result in sin, while still maintaining innocence until a sin is actually committed by free will. The Arminian John Miley in his book Systematic Theology attempts to explain this point regarding article VII, lest we confuse the idea with Calvinism:

There is not one word about a sharing of the race in the sin of Adam, or about the corruption of human nature as a judicial infliction on the ground of a common Adamic guilt. Nor is there one word which expresses or even implies an intrinsic sinfulness and damnableness of this inherited corruption of nature. Therefore we could controvert these special elements of the Augustinian doctrine, as we have done, without the slightest departure from our own doctrine as formulated in this article. (John Miley's Systematic Theology, p523)

Note: I am not sure Miley truly represents original Methodist though under John Wesley, but am pretty sure is he correctly representing original Remonstrant beliefs, which seem to also be his own.

This fully confirmed by own understanding of the difference. Therefore to answer the question: In simple terms the remonstrants replace Augustinian/Calvinistic election with freedom of will. Therefore a person was not born guilty with a sinful nature but born innocent. Being born innocent does not mean that a person can do good but just means they are not personally guilty and made sinful by participating in Adam's sin and punishment for it.

The third article regarding saving faith is simply to say without faith one can't be saved or do good. What is NOT said is that humans can't do good because before they were born they were damned into an actual wicked nature and that as soon as their souls ate capable of expressing the slightest of moral inclination or choice they will sin accordingly.

The fourth article just assigns grace as the procuring cause of anything that a man can be enabled to do that is good. Then it is affirmed that grace is not provided to the elect but that all men are provided grace so that they can receive or not receive Christ. The directive of both articles is to establish two things that lay the foundation of all Arminian thought.

1. Faith and grace are not denied when departing from traditional Augustinian/Calvinistic views of original sin.
2. Human freedom, guilt and moral depravity can only be counted from one's own choice and not imputed from Adam in the form of personal guilt and corresponding actual perverseness in birth.

Arminian doctrine as defined by Arminias is largely defined by how it disagrees with Calvinism and these two articles establish this position by showing what they retain and what they do not say.

Reference

One book that deals heavily with the difference between Airman concepts of original sin and Calvinist concepts is from John Owen entitled 'A display of Arminianism' in the Works of John Owen Volume VII.

Owen begins by charging Arminians with overthrowing all traditional beliefs on the subject. (Roman Catholic dogma on this area is actually very close to Calvin as well by the way at least compared to Arminian).

My purpose for the present is not to allege any testimonies of this kind; but, holding myself close to my first intention, to show how far in this article, as well as others, the Arminians have apostated from the pure doctrine of the word of God, the consent of orthodox divines, and the confession of this church of England. In the ninth article of our church, which is concerning original sin, I observe especially four things:—First, That it is an inherent evil, the fault and corruption of the nature of every man. Secondly, That it is a thing not subject or conformable to the law of God, but hath in itself, even after baptism, the nature of sin. Thirdly, That by it we are averse from God, and inclined to all manner of evil. Fourthly, That it deserveth God’s wrath and damnation. All which are frequently and evidently taught in the word of God, and every one denied by the Arminians, as it may appear by these instances, in some of them:— First, That it is an inherent sin and pollution of nature, having a proper guilt of its own, making us responsible to the wrath of God, and not a bare imputation of another’s fault to us his posterity: which, because it would reflect upon us all with a charge of a native imbecility and insufficiency to good, is by these self-idolizers quite exploded. 1“Infants are simply in that estate in which Adam was before his fall,” saith Venator.2 “Neither is it at all considerable whether they be the children of believers or of heathens and infidels; for infants, as infants, have all the same innocency,” say they jointly, in their Apology: nay, more plainly,3 “It can be no fault wherewith we are born.” In which last expression these bold innovators, with one dash of their pens, have quite overthrown a sacred verity, an apostolic, catholic, fundamental article of Christian religion. (Works of John Owen Volume VII)

After mentioning various proofs of the Calvinistic doctrine of original sin, Owen goes on to explain how Adam's sin imputed to human nature is not just a bare imputation of another’s fault, but an intrinsically inner corruption of our nature and shared guilt. That is to say that we are guilty of sin before we are born and we crave sin as soon as we are possible to crave moral things. Then he explains that Arminian thoughts that oppose this traditional concept:

The Arminians deny all such imputation, as too heavy a charge for the pure, unblamable condition wherein they are brought into this world. They deny, I say, that they are guilty of Adam’s sin, as sinning in him, or that his sin is any way imputed unto us; which is their second assault upon the truth of this article of faith. 1“Adam sinned in his own proper person, and there is no reason why God should impute that sin of his unto infants,” saith Boræus. The nature of the first covenant, the right and power d God, the comparison instituted by the apostle between Adam and Christ, the divine constitution, whereby Adam was appointed to be the head, fountain, and origin of all human kind, are with him no reasons at all to persuade it.2 “For it is against equity,” saith their Apology, “that one should be accounted guilty for a sin that is not his own,—that he should be reputed nocent who, in regard of his own will, is truly innocent.” And here, Christian reader, behold plain Pelagianism obtruded on us without either welt3 or guard; men on sudden made pure and truly innocent, notwithstanding all that natural pollution and corruption the Scripture everywhere proclaims then to be replenished withal. Neither is the reason they intimate of any value, that their wills assented not to it, and which a little before they plainly urge. “It is,” say they,1 “against the nature of sin that that should be counted a sin to any by whose own proper will it was not committed:” which being all they have to say, they repeat it over and over in this case,—“It must be voluntary, or it is no sin.” But I say this is of no force at all; for,—first, St John, in his most exact definition of sin, requires not voluntariness to the nature of it, but only an obliquity, a deviation from the rule. It is an anomy,—a discrepancy from the law, which whether voluntary or no it skills not much; but sure enough there is in our nature such a repugnancy to the law of God. So that, secondly, if originally we are free from a voluntary actual transgression, yet we are not from an habitual voluntary digression and exorbitancy from the law. But, thirdly, in respect of our wills, we are not thus innocent neither; for we all sinned in Adam, as the apostle affirmeth. Now, all sin is voluntary, say the Remonstrants, and therefore Adam’s transgression was our voluntary sin also, and that in divers respects..... (Works of John Owen Volume VII)