My first inclination is sure, allow it. But...
change the story a little: a player decides to make a character who is missing their left hand. Again they don't ask anything in return. Would I allow them, in a moment of need, to suddenly regenerate their hand? No, I don't think I would. That's just not the way hands work.
The aasimar thing is different, we don't have a built-in expectation for how aasimar wings work. Is that enough different? I'm not really sure. It probably depends on your campaign's concept of aasimar.
(But also: is there really a winged aasimar variant in SCAG? I didn't think there was.)
Personally, I think the difference is mechanics versus story, or crunch versus fluff.
If a player wants a character that is missing a hand, that doesn't necessarily have an impact on mechanics. Such a character still has the ability to use shields, two handed weapons, cast spells with somatic components, etc. They may find different ways to hold things, ways to use their stump, or use some kind of graft or prosthetic, but that is only limited by one's imagination. Mechanically, as an example, the only thing that wielding a 2-handed weapon really means, at its core, is that you could not wield this weapon and benefit from a shield or cast spells with somatic components while using it. While the name its given implies needing two hands, that's really just a shorthand way of limiting what you can do while wielding a particular type of weapon that is more damaging than other types.
Following this, a one handed character is no more limited than a two handed character.
That is, until you apply a concrete penalty or mechanical consequence for that character choice. This most often occurs as the result of an injury in game. A character doesn't lose a hand without a mechanical consequence.
But a player could create a character that has only one hand and has no mechanical consequence or limitation. Of course, in this situation, the impact is more on the story/RP/fluff. Such a character could not describe using their stump like a hand, but they could imagine ways to get around that problem. As long as there is no mechanical implication or change, there is effectively no difference.