This, like many topics of this nature, seems like the kind of thing that should be addressed before the first session. It's not like most games have actual mechanics for dismemberment, so it's something the GM should have in mind as a possibility before they even get started, and that shouldn't be a possibility that they should keep to themselves.
In my experience (somewhat extensive) somewhere around 1/4 of RPGs have rules for some form of lasting injury; most of those it's an optional rule.
I never
ask about it as a GM; I do
warn about it in systems where it exists. By lasting, I mean something that isn't just "some down time and it's over"...
Some I've run where it's core: L5R 5E, Warhammer FRP (1, 2, 3, and while I've not run it, 4), RuneQuest 3e, Elf Quest, Worlds Beyond, FFG Star Wars, Alien (YZE), Vaesen (YZE) Twilight 2000 (1e, 2e, 2.2e, 4e), Dark Conspiracy (1e), Spirit of the Century (Fate system), Dresden Files (1st ed), Hârnmaster, Rhand (LEG), Aliens Adventure Game (LEG), GURPS, Rolemaster, Spacemaster (1E and 2E),
A few I've not run that do have lasting wounds as part of core mechancs: Fate Core, FFG's End of the World line, Genesys (Same mechanics as FFG Star Wars),
Hmmm.... most of those do so via critical hit rules. GURPS, BRP (RQ, EQ and WB) by excess damage to hit location. Fate (SotC,Fate Core, Dresden Files) and EotW do so by trading in harm for a lasting injury, freeing up damage space. Fate, that can eventually be bought off.
The crit table types (L5R5e, WFRP all, T2K4, all the FFG games, HM, both LEG games) include results removing limbs.
That list above represents about 5% of the games I've run over the last 40 years in terms of systems/editions; it represents closer to 40% of my game time...
A few I'm not able to check at the moment:
BTRC's CORPS & EABA ( I recall the broken bones category)
Palladium's games (all have a provision for it; I don't recall if it's optional or core - when taking HP damage instead of SDC, roll for lasting injury on a table.)
that's about 30 editions of arouond 400 editions/games I've played or run over the last 40 years. So about 8%. Many more have an optional rule for it in core; more still added it in a subsequent publication.