OK, so Aristotle doesn't really take things definitively one way or the other - he denies pangenesis, allows for the idea of acquired characteristics being inherited and how that would be explained within his theory, but doesn't argue strongly that this is actually the case either. So let's set aside Aristotle for the time being.
If we look at some other people who might be considered authorities in 1220 (or who will soon become authorities!) pangenesis, and by extension the inheritance of acquired traits, is widely supported. The quotes below are not my own research (who would have time for that?) but instead are drawn from this article on the "History of the Idea of the Inheritance of Acquired Characters and of Pangenesis":
https://www.jstor.org/stable/1005592
However for the benefit of people without access, I've re-quoted some of the excerpts below. Generally the idea of pangenesis is by far the most accepted view. Aristotle is for the most part alone in his refutation of it (at least among those authorities who were transmitted and were widely referenced in the 13th century), and almost no one denies the inhertiance of acquired traits, even if they reject pangenesis. It seems to have been mostly an accepted fact. Most importantly, it's supported by the leading minds of the 13th century (well, those who weigh in on the subject, anyway).
This for me is sufficient to conclude that both the inheritance of acquired traits and pangenesis are likely to be true in Mythic Europe. But this is only my opinion and as with all these things it's fundamentally up to the individual saga to say what's really what in Mythic Europe.
By Hippocrates (in The Sacred Disease) and other followers of the Hippocratic tradition:
As the seed comes from all parts of the body, healthy particles will come
from healthy parts, and unhealthy from unhealthy
and in Airs, Waters, and Places:
I will begin with the Longheads.. There is no other race at all with heads like theirs. Originally custom was chiefly responsible for the length of the head, but now custom is reinforced by nature. Those that have the longest heads they consider the noblest, and their custom is as follows. As
soon as a child is born they remodel its head with their hands, while it is still soft and the body tender, and force it to increase in length by applying bandages and suitable appliances, which spoil the roundness of the head and increase its length. Custom originally so acted that through force such a nature came into being; but as time went on the process became natural, so that custom no longer exercized compulsion. For the seed comes from all parts of the body, healthy seed from healthy parts, diseased seed from diseased parts. If, therefore, bald parents for the most part bald children, grey-eyed parents grey-eyed children, squinting parents squinting children, and so on with other physical peculiarities, what prevents a longheaded parent having a long-headed child?
Galen possibly followed in the tradition of Hippocrates, though with less certainty. The quote here is actually from Albertus Magnus summarising the Galenic position, I couldn't find the relevant quote directly from Galen, and the Galenic corpus is riddled with misatributions anyway but at least Albertus Magnus believed this was Galen's position:
Galen said however that he didn't know whether what Hippocrates says was true or not: but he said á propos of this idea that it was more probable that the sperm, which was the "'superfluous'" of the fourth digestion, exudes from all members of the body but especially from the head and receives by chance a mixture in the head for it is completed there more quickly, because in the head are the more noble forces of the soul.
Bartholomaeus Anglicus (just after 1220):
The matere of ye chyld is matere semniialis, that is sliedde by werkyng of generacon
And comyth of all ye parties of the fader & the moder
Albertus Magnus himself in De Animalibus book 3:
here is proof that [sperm] exudes from all members of the body because it has the potentiality of forming the whole body: and we see in many animals a member lacking at birth which was deficient in the generating forces.
and in Animalia book 15:
Besides, if what they say is true, the sons of those having defects will have imperfect and diminished members; always they will be with imperfection; and they will be incomplete and diminished in their members and this defect we see to exist with our own eyes. Nevertheless, in the following writings, we shall inquire the cause of both of these- namely, as much the cause of the similarity of children with their parents as the reason of the diminution of the members, which exists sometimes in parents and not in sons. For this question concerning the aforesaid accidents of both [similarity and diminution] is a common one.
He also has a rather long passage in De Nutrimento book 1 about it, but it's mainly refuting specific details of how the sperm is mixed and formed so it's not worth quoting here.
Thomas Aquinas also has a rather lengthy essay on the topic which comes to roughly similar (pro-pangenesis) conclusions. And specifically on the topic of the inheritance of acquired traits he writes (most concerned, as always, with the theological implications):
...thus a leper may beget a leper, or a gouty man may be the father of a gouty son, on account of some seminal corruption, although this corruption is not leprousy or gout... But all these explanations are insufficient. Because, granted that some bodily defects are transmitted by way of origin from parent to child, and granted that even some defects of the soul are transmitted in consequence, on account of a defect in the bodily habit, as in the case of idiots begetting idiots; nevertheless the fact of having a defect, by way of origin seems to exclude the notion of guilt, which is essentially something voluntary.
And by Roger Bacon (Opus Majus, part 6), mainly concerned with why men seem to live shorter lives than the biblical patriarchs:
Very rarely does it happen that anyone pays sufficient heed to the rules of health. No one does so in his youth, but sometimes one in three thousand thinks of these matters when he is old and approaching death, for at times he fears for himself and thinks of his health. But he can- not then apply a remedy because of his weakened powers and sense and his lack of experience. Therefore fathers are weakened and beget weak sons with a liability to premature death. Then by neglect of the rules of health the sons weaken themselves, and thus the son's son has a doubly weakened constitution, and in his turn weakens himself by a disregard of these rules. Thus a weakened constitution passes from father to sons, until a final shortening of life has been reached as is the case in these days ... the longevity of man has been shortened contrary to nature. Moreover, it has been proved that this excessive shortening of the span of life has been retarded in many cases, and longevity prolonged for many years by secret experiment.
The last part of which sounds suspiciously like he's talking about magi...
Whoof, that ended up a wall of text. On the bright side, writing this has been giving me all sorts of ideas about how this could be applied for original research on medical magic...