Were I running 5e Drow might well be* kill-on-sight there too; as would demons, devils, jellies, oozes, githi, chromatic dragons, giants, and various other monsters.
* - almost certainly would be, actually, given that most of the other usual humanoid kill-on-sight options have been neutered into blandness.
Then you would be running 5e in a way specifically not intended, which contradicts the explicit text given to players. 5e made very extra explicit what had been a strong implication of 4e and a quiet whisper of 3e: sapient races that aren't monsters can't be 100% guaranteed Always Chaotic Evil (or whatever), and specifically with Drow, they're now explicitly understood to encompass a spectrum of cultures--some of which are devotees of Lolth (or other evil Drow deities), some of which are not (possibly devotees of good Drow deities, like Eilistraee, but possibly not having any special religious affiliation). More or less, it's saying that for established settings (such as FR, the implied setting of 5e), the Drow everyone is familiar with come from only a handful of cities in a particular region--with other regions having different cultures. More or less, it would be like if the only two nations of humans one were exposed to were Thay and the Zhentarim, and thus asserting that the entire human race must be KOS-level Pure Evil because of it.
Now, of course, you aren't beholden to what WotC decided to do with the game. But the fact is, Drow are explicitly an entirely playable race/species in
both the 2014 PHB and 2014 PHB. Even if they weren't, the mechanical package doesn't change other than which "you can cast this once a day" extra spells (and otherwise-ordinary cantrip) you get, so for almost all rules purposes, a drow is just an elf with dark skin and slightly different bonus spells. Telling the player they can't play an Elf in general, Drow or not, would almost surely get some pushback.
And yes,
part of it is Drizzt Do'Urden--but part of it is also just that genuinely sapient species/races are being treated as "it's more complex than that" now.
Some of them, from particular factions, may be "show no mercy". But you can't just instantly see the color of an elf's skin and know "oh this is 100% an un-person I need to kill right away." For, I should hope,
probably obvious reasons. (Social implications from demonizing a dark-skinned, matriarchal society that lives in a dangerous faraway place?
Perish the thought!)