You know what I have a problem with? That a player who wants to play a Drizzt clone thinks the DM should throw out all logic, reason and established campaign lore. While I don't remember the last time I used drow in my campaign they are the bogeyman. The monster that comes in the night, burns the village to the ground and disappears without a trace. Survivors are extremely rare but tell of coal black figures in the night torturing and killing anyone who fought or was too old or lame. The rest were taken as prisoners never to be seen again.
So a drow that tried to walk into a city? Sorry, they'd be killed on sight. That might not be a good thing, it might not be the right thing, but it would be the realistic thing. I do my best to base my world on logic, history and what I think would actually happen. People react to imminent threats with a fight or flight response.
If I made up a new world for every player and every campaign maybe it would be different, maybe not*. But honestly? I'd probably respond the same as the guy that wanted to run a half-dragon half-vampire. You can always ask, we can always discuss, but sometimes the answer is no.
*I don't have problems with monsters, even some that look vaguely human but I'm not getting into that argument again.
Thank you for reminding me of another thing I hate and loathe: Monocultures. The "All X are Y" side of things. Which is not, y'know, realistic in the slightest.
How are drow in your world always 'The boogyman'? Do they not have people seeking ambition by other means? Perhaps, say, money? Selling secrets off to the surface folk so one of their enemies gets taken down a few pegs? Or, heck, drow who want to see more than what's there? Wizards fed up with the politics and instead deciding that only by seeking out all arcane knowledge can they be content? Or, heck, rebels and the ever popular Dritz archetype, which is popular for a heck of a lot of reasons and unquestionably a strong archetype?
Don't forget the last time D&D had a really strong archetype counter to the established race, they scrapped and rewrote the whole race, and that's why Githzerai have a niche today rather than being 'i unno githyanki but less naughty words lol'. Popular don't mean bad, and honestly, the strength of the Dritz archetype is probably salvaging part of the flanderised mess that Drow were earlier in FR lifespans. The Dritz archetype made drow worth playing for people because it was a logical outcome to the Drow problem of 'Why would you actually want to live in a civilisation like this and not burn every single bridge the moment you can and go somewhere that's actually, y'know, reasonable to live in'
Logically and reasonably there has to be divide in a society because, struth, even eusocial things like ants have personality. Drow being only raiders and never anything more than that is not realistic, plain and simple.
Shooting an unknown person walking into a city when there have been no firm reports of them, per your own words, is a bloody reckless action. Arresting for questioning? Sure, that's sensible, reasonable, because you don't know who they are, what they're doing, or who they represent. Per the campaign lore you've presented here, a guard who shoots someone like this walking into a city? I'd expect their ass to be arrested because, well, can't very well interrogate the dead as to why this is happening, or where people have been taken to. Can't discover there's fractures in the drow society to exploit and gain a foothold to stop a problem if you do that, can you?
I dislike its wider thing as well. "All forest elves have this one culture, no matter where they live" is just as bad as "All drow are just drow stereotypes with no divergence from the normal". But, I consider adventuring and just meeting and talking and learning from other places to be a big thing in the D&D experience. Less Mos Eisley Cantina where stuff is just fancy to catch the eye, and more Morrowind or Final Fantasy, where you're thrown full on into a culture you don't understand and just have to adapt, learn, and get respect