I couldn't find any quotes concerning the Church being a "hospital for sinners" (or a variation) among the works of the Fathers I checked (nor could I find a reference/address for the St. John Chrysostom quote you mentioned), but fwiw there is a passage in St. Gregory the Theologian about how priests act as a "physician of souls"...
But granted that a man is free from vice, and has reached the greatest heights of virtue: I do not see what knowledge or power would justify him in venturing upon this office. For the guiding of man, the most variable and manifold of creatures, seems to me in very deed to be the art of arts and science of sciences. Any one may recognize this, by comparing the work of the physician of souls with the treatment of the body; and noticing that, laborious as the latter is, ours is more laborious, and of more consequence, from the nature of its subject matter, the power of its science, and the object of its exercise. The one labours about bodies, and perishable failing matter, which absolutely must be dissolved and undergo its fate, (Gen. 3:19) even if upon this occasion by the aid of art it can surmount the disturbance within itself, being dissolved by disease or time in submission to the law of nature, since it cannot rise above its own limitations.
The other is concerned with the soul, which comes from God and is divine, and partakes of the heavenly nobility, and presses on to it, even if it be bound to an inferior nature. Perhaps indeed there are other reasons also for this, which only God, Who bound them together, and those who are instructed by God in such mysteries, can know, but as far as I, and men like myself can perceive, there are two: one, that it may inherit the glory above by means of a struggle and wrestling (Eph. 6:12) with things below, being tried as gold in the fire (1 Pet. 1:7) by things here, and gain the objects of our hope as a prize of virtue, and not merely as the gift of God. This, indeed, was the will of Supreme Goodness, to make the good even our own, not only because sown in our nature, but because cultivated by our own choice, and by the motions of our will, free to act in either direction. The second reason is, that it may draw to itself and raise to heaven the lower nature, by gradually freeing it from its grossness, in order that the soul may be to the body what God is to the soul, itself leading on the matter which ministers to it, and uniting it, as its fellow-servant, to God.
Place and time and age and season and the like are the subjects of a physician's scrutiny; he will prescribe medicines and diet, and guard against things injurious, that the desires of the sick may not be a hindrance to his art. Sometimes, and in certain cases, he will make use of the cautery or the knife or the severer remedies; but none of these, laborious and hard as they may seem, is so difficult as the diagnosis and cure of our habits, passions, lives, wills, and whatever else is within us, by banishing from our compound nature everything brutal and fierce, and introducing and establishing in their stead what is gentle and dear to God, and arbitrating fairly between soul and body; not allowing the superior to be overpowered by the inferior, which would be the greatest injustice; but subjecting to the ruling and leading power that which naturally takes the second place: as indeed the divine law enjoins, which is most excellently imposed on His whole creation, whether visible or beyond our ken.
This further point does not escape me, that the nature of all these objects of the watchfulness of the physician remains the same, and does not evolve out of itself any crafty opposition, or contrivance hostile to the appliances of his art, nay, it is rather the treatment which modifies its subject matter, except where some slight insubordination occurs on the part of the patient, which it is not difficult to prevent or restrain. But in our case, human prudence and selfishness, and the want of training and inclination to yield ready submission are a very great obstacle to advance in virtue, amounting almost to an armed resistance to those who are wishful to help us. And the very eagerness with which we should lay bare our sickness to our spiritual physicians, we employ in avoiding this treatment, and show our bravery by struggling against what is for our own interest, our skill in shunning what is for our health.
For we either hide away our sin, cloaking it over in the depth of our soul, like some festering and malignant disease, as if by escaping the notice of men we could escape the mighty eye of God and justice. Or else we allege excuses in our sins, by devising pleas in defence of our falls, or tightly closing our ears, like the deaf adder that stops her ears, we are obstinate in refusing to hear the voice of the charmer, and be treated with the medicines of wisdom, by which spiritual sickness is healed. Or, lastly, those of us who are most daring and self-willed shamelessly brazen out our sin before those who would heal it, marching with bared head, as the saying is, into all kinds of transgression. O what madness, if there be no term more fitting for this state of mind! Those whom we ought to love as our benefactors we keep off, as if they were our enemies, hating those who reprove in the gates, and abhorring the righteous word; (Amos 5:10) and we think that we shall succeed in the war that we are waging against those who are well disposed to us by doing ourselves all the harm we can, like men who imagine they are consuming the flesh of others when they are really fastening upon their own.
For these reasons I allege that our office as physicians far exceeds in toilsomeness, and consequently in worth, that which is confined to the body; and further, because the latter is mainly concerned with the surface, and only in a slight degree investigates the causes which are deeply hidden. But the whole of our treatment and exertion is concerned with the hidden man of the heart, (1 Pet. 3:4) and our warfare is directed against that adversary and foe within us, who uses ourselves as his weapons against ourselves, and, most fearful of all, hands us over to the death of sin. In opposition then, to these foes we are in need of great and perfect faith, and of still greater co-operation on the part of God, and, as I am persuaded, of no slight countermanoeuvring on our own part, which must manifest itself both in word and deed, if ourselves, the most precious possession we have, are to be duly tended and cleansed and made as deserving as possible.
To turn however to the ends in view in each of these forms of healing, for this point is still left to be considered, the one preserves, if it already exists, the health and good habit of the flesh, or if absent, recalls it; though it is not yet clear whether or not these will be for the advantage of those who possess them, since their opposites very often confer a greater benefit on those who have them, just as poverty and wealth, renown or disgrace, a low or brilliant position, and all other circumstances, which are naturally indifferent, and do not incline in one direction more than in another, produce a good or bad effect according to the will of, and the manner in which they are used by the persons who experience them. But the scope of our art is to provide the soul with wings, to rescue it from the world and give it to God, and to watch over that which is in His image, (Gen. 1:26) if it abides, to take it by the hand, if it is in danger, or restore it, if ruined, to make Christ to dwell in the heart (Eph. 3:17) by the Spirit: and, in short, to deify, and bestow heavenly bliss upon, one who belongs to the heavenly host.
-- St. Gregory the Theologian, Oration 2.16-22