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.
The way I've always had it, Llolth killed off any other deity the Drow ever tried to raise against her, thus ensuring she's the only in-culture one they've got. Ditto with Gruumsh and Orcs.
A Drow could, of course, go out-of-culture to find a deity, but that would pretty much amount to self-exile.
Now, of course, you aren't beholden to what WotC decided to do with the game.
Oh, I most certainly am not.
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.
Same as Dwarves are PC-playable but Duergar are not, or Gnomes are PC-playable while Svirfneblin are not, Elves are PC-playable while Drow are not. Each of the non-PC species are the "underdark mirror" of their surface equivalents.
That said, if the game was set full-time in the underdark I'd be tempted to reverse all that.
Side question - and this came up at our last session - what's the underdark mirror for Hobbits?
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!)
I get that for Drow in particular, which is part of the reason they're pretty much gone from my current setting. But other monsters IMO should be just that: monsters, until and unless an individual proves itself otherwise.
Orcs, for example, are usually kill-on-sight in my game, but tell that to the party who ended up taking surrender from some Orcs then taking those surrenderees in as paid party henches. Those Orcs turned out to be loyal to a fault - all they needed was someone to treat 'em right - to the point where the party Ranger (who mechanically gets combat bonuses against Orcs) ended up leaving the party to join the Orcs and become their liaison with the neighbouring Humans.