Regeneration: I like that the hand healing was agonizing. I remember reading a bit about surgery pre anesthesia. Amputations were done super fast to minimize the pain. Mastectomies could not be done fast and were mind-bendingly prolonged, intense agony. Not uncommon for surgery to be rejected because the pain level would be unbearable.
So, imagined losing your hand to not one cut, but rather having it repeatedly sliced off from the fingertips down to the wrist. Not only did Serina have to go through that (in reverse), but she had to do it to herself, over the course of (probably?) multiple days.
And Almelia had to agree to that, knowing that it was what Serina wanted, and that Serina was doing it in part for Amelia. Then Almelia sits through it by Serina's side not able to do much other than hold Serina's hand and support her.
Part of what I like here is the contrast between the generally light hearted nature of the story, and the sudden grim reality of Serina losing her hand and agony of its restoration.
As for "asspull"... that's ridiculous. Serina does know more than about how human body works, and she has a unique ability to control magic at a very fine level. And frankly, it's much less of an asspull than magic mechanical hand. (Just because we're used to the latter from other stories doesn't make them less absurd.)
As for impact on the world... Hard to say... Healing here is a combination of Serina's insanely (and uniquely) strong control of magic and her knowledge of the finer structures in biology. Even if she can't really explain the biology, she could at least get them started on microscopy, which could lead to some improvements in their biology knowledge. But... It's also possible that her "genes" explanation gets dismissed as fantasy by the healing experts of the time -- overturning well established knowledge is always difficult.
However even if experts learn of genes, that might not help their healing magic if such magic requires Serina's crazy fine control.
If Serina can use that magic on other people, then she could/should suddenly gain "miracle doctor" status. But author can avoid that by having Serina's healing magic only work on herself.
Prince... His reaction is somewhat understandable, because what Serina just claimed is considered impossible. (Same as if someone claimed to have regrown their whole hand in our world.)
OTOH, he witnessed Serina's severed hand himself. OTOOH, the reasonable conclusion on his part (since regrowth is impossible) is that the loss of her hand was the work of illusion magic or some other fakery.
But clearly he is not being rational, but is emotionally driven here. He loathes the appearance of failure -- and his friendly fire was a massive screwup. Just straight incompetence demonstrating unfitness to rule.
He's torn between not wanting to admit to any incompetence, and the desire/necessity of doing the right thing. Then when he bends/compromises enough to pay some recompense, he's presented with a situation where it looks like the whole thing was a fraud by the woman who already upstaged and embarrassed him.
(BTW, it would make sense for there to be additional testimony regarding the healing (multiple witnesses of loss of hand; other people witnessing the days long, agonizing regeneration process. OTOH, Serina and Almelia might have intentionally limited witnesses since then news of the incident could be suppressed and thus not embarrass the prince.)
Serina & Almelia: Serina is airheaded and oblivious. But hopefully Almelia's apology is enough for Serina to begin to grasp that Almelia really does care deeply for her. Serina likes Almelia, and maybe cares more for Almelia than she herself knows. But it's still a huge jump from that to mutual committed romantic love.