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I don't see how Anne Catherine Emmerich is contradicting Enoch's story, whom she praises as a holy man.  It's unclear what she means when she says "when the angels fell".  Is she talking about satan's revolt against God vs St Michael?  I don't see how because she says they won't be cast into hell before the last day.  But satan and his angels are already in hell.  So she must be talking about fallen angels in the same way that Enoch was, which means, angels fell a 2nd time.  She uses the term (assuming the translation is accurate) of "fallen spirits".  But satan and co are not fallen spirits but demons who are in hell.  So, it appears that the "sons of god" does mean actual angels.

I have frequently seen that, when the angels fell, a certain number had a moment of repentance and did not in consequence fall as low as the others.

Much of the argumentation seems to come (at least later, by the time of the scholastics) from this notion that angels are pure spirit.  Where is that taught or to be found in Sacred Scripture?  Whence does that notion originate?  I haven't seen any authority for that supposition, whether from Scripture or from Tradition.  And yet it seems to be simply assumed, and then they draw conclusions from that presupposition.  But what if it's not the whole story.  Certainly it's clear that there are some higher choirs of angels that are indeed pure spirit.  But why the different choirs and the hierarchy.  To what extent are the lower choirs "inferior" to the higher ones?  What is there that rules out the existence of a lowest choir where those creatures of God had spirit plus a sensible nature along with the ability to take on bodies (although not of earth like man but of a more aetherial substance).

In any case, we tend to lump all these creatures into the category of "angel", but there were differences and degrees of sublimity in the hierarchy from God down?  It could be circular reasoning, where you define "angel" as a being of pure spirit, and then any that are called angels are supposed to be in this same category.  In point of fact, I believe St. Thomas and others held that you can't really classify angels that way, in that every angel is his own special category and has a unique nature.

It would be interesting to dig up anything that the Fathers had to say about angels in general.

In any case, the very etymology of angel simply means "messenger" and by itself the term does  not define them to be pure spirits, just as these transitional creatures between God, the invisible creation, and visible creation.

On the contrary, It is said (Ps 103:4): Who makes His angels spirits.

I answer that, There must be some incorporeal creatures. For what is principally intended by God in creatures is good, and this consists in assimilation to God Himself. And the perfect assimilation of an effect to a cause is accomplished when the effect imitates the cause according to that whereby the cause produces the effect; as heat makes heat. Now, God produces the creature by His intellect and will (Q. 14, A. 8; Q. 19, A. 4). Hence the perfection of the universe requires that there should be intellectual creatures. Now intelligence cannot be the action of a body, nor of any corporeal faculty; for every body is limited to here and now. Hence the perfection of the universe requires the existence of an incorporeal creature.

The ancients, however, not properly realizing the force of intelligence, and failing to make a proper distinction between sense and intellect, thought that nothing existed in the world but what could be apprehended by sense and imagination. And because bodies alone fall under imagination, they supposed that no being existed except bodies, as the Philosopher observes (Phys. iv, 52,57). Thence came the error of the Sadducees, who said there was no spirit (Acts 23:8).

But the very fact that intellect is above sense is a reasonable proof that there are some incorporeal things comprehensible by the intellect alone.
https://aquinas.cc/la/en/~ST.I.Q50.A1.SC

On the contrary is the authority of Holy Scripture wherein they are so named. For the name Seraphim is found in Isa. 6:2; the name Cherubim in Ezech. 1 (Cf. 10:15,20); Thrones in Col. 1:16; Dominations, Virtues, Powers, and Principalities are mentioned in Eph. 1:21; the name Archangels in the canonical epistle of St. Jude (9), and the name Angels is found in many places of Scripture.

I answer that, As Dionysius says (Coel. Hier. vii), in the names of the angelic orders it is necessary to observe that the proper name of each order expresses its property. Now to see what is the property of each order, we must consider that in coordinated things, something may be found in a threefold manner: by way of property, by way of excess, and by way of participation. A thing is said to be in another by way of property, if it is adequate and proportionate to its nature: by excess when an attribute is less than that to which it is attributed, but is possessed thereby in an eminent manner, as we have stated (Q. 13, A. 2) concerning all the names which are attributed to God: by participation, when an attribute is possessed by something not fully but partially; thus holy men are called gods by participation. Therefore, if anything is to be called by a name designating its property, it ought not to be named from what it participates imperfectly, nor from that which it possesses in excess, but from that which is adequate thereto; as, for instance, when we wish properly to name a man, we should call him a rational substance, but not an intellectual substance, which latter is the proper name of an angel; because simple intelligence belongs to an angel as a property, and to man by participation; nor do we call him a sensible substance, which is the proper name of a brute; because sense is less than the property of a man, and belongs to man in a more excellent way than to other animals.

So we must consider that in the angelic orders all spiritual perfections are common to all the angels, and that they are all more excellently in the superior than in the inferior angels. Further, as in these perfections there are grades, the superior perfection belongs to the superior order as its property, whereas it belongs to the inferior by participation; and conversely the inferior perfection belongs to the inferior order as its property, and to the superior by way of excess; and thus the superior order is denominated from the superior perfection.

So in this way Dionysius (Coel. Hier. vii) explains the names of the orders accordingly as they befit the spiritual perfections they signify. Gregory, on the other hand, in expounding these names (Hom. xxxiv in Evang.) seems to regard more the exterior ministrations; for he says that angels are so called as announcing the least things; and the archangels in the greatest; by the virtues miracles are wrought; by the powers hostile powers are repulsed; and the principalities preside over the good spirits themselves.

https://aquinas.cc/la/en/~ST.I.Q108.A5.SC