$250 can go a very long ways if you use PDFs. Obviously, games with lots of supplements like D&D or Pathfinder can get expensive.

Honestly, I would just ask your players what they’re interested in, genre-wise and complexity-wise, before committing to any purchases.

I will always recommend Lancer!

It’s a mecha-themed tabletop game with tons of customization and a great emphasis on battle strategy. The combat is super fun and works a lot like a tactics video game such as XCOM. The out of combat gameplay is lighter than most, but still has plenty of stuff to customize and do. The game’s lore is well-written and actually interesting, and a lot of fun to look through (but ignorable if you want to customize, of course). The game also has a ton of free modules and stuff online such as Comp/Con that allows you to customize characters and mechs easily and for free. In fact, the game as a whole is very cheap, and only requires the GM to purchase the books.

Best of all, the Devs are amazing (if they aren’t leftists, they’re very close) and the community is incredibly wholesome and is very LGBTQ friendly. Lots and lots of trans comrades are always talking about the game and stuff on the official Discord. The Discord also has a million free resources like maps and art and all that to make GMing that much easier.

If you want to hold campaigns where trans mech pilots destroy capitalists with heavy weaponry and eldritch-adjacent superweapons, Lancer has you covered!

Fun fact: The entire rules for Pathfinder (both 1E and 2E) are on this website

The only things not present are flavor bits, world information and the pre-written adventures

So, you could get pretty far without spending a single dollar

Well first of all, if you’re OK with PDFs then you can get a lot of things for free you know?

Pathfinder 2e is, IMO, a better and very fun version of dnd, its a good run-of-the-mill d20 fantasy system with a setting that I enjoy quite a lot with increasing queer writing in it. And ALL the rules are free and easily found and used by going to 2e.aonprd.com/, the official PRD. Yes, aallllll the rules, updated suite quickly when new stuff comes out. Foundry pf2e has great support and very nice to use, and pathbuilder is a godsend for character creation.

There are so many cool and interesting systems to try out there and really you should try them, a lot of them are simple and easy to setup and you can easily get a one shot going to see if you want to keep going.

If you need any PF1 or PF2 PDFs, imu. And in general of you need any ttrpg PDFs of any kind, I’m pretty good at finding them (except the tiny Indy stuff that gets crowd funded, sometimes you can find it but its much harder)

Fantasy Flight Star Wars is good for Star Wars
FATE is general purpose, good for anything narrative heavy
there are many i could recommend given the time lol

Pathfinder is free

Beam Saber looks pretty cool if you and/or the friends you are playing with like giant robots and sci fi anime.

You could go for Call of Cthulhu or its more action-oriented offshoot Delta Green. That one is like Men in Black or X-Files. There’s also plenty of cheap and free modules to run with for that. Another Option is Savage Lands. That’s a very barebones system that can be applied to anything, much like GURPS. But unlike GURPS, it’s super basic in its rules and relies healthily on cheat tokens. If you go with Savage Lands, buy it online since it comes with a PDF code if you buy it off the webstore. With the basebook and some Deadlands materials, you could run a really simple, but exciting Southern Gothic action-adventure for under $100.

First things first, try to pirate everything you can from here before you buy anything (fair warning: it’s from 4chan but it’s too good a source to pass up). Since you seem to be fine with any of fantasy, sci-fi, or historical games, I think the two most important questions are how willing/able you and, more importantly, your players are to read rules/learn complicated systems, and what kind of gameplay the group interested in. A lot of the things you list off as likes are more things that you would build into the setting during your planning, where the system itself won’t necessarily matter.

• Pathfinder/D&D is super combat heavy and generally complex. You can run these in non-european settings with some work (PF has a few regions that are fleshed out IIRC).
• Call of Cthulhu is a lot less focused on concrete rules and focuses on investigation into things beyond mortal comprehension, with combat usually resulting in horrible death because the party are rando investigators and not professional soldiers. Set in the 1930s, could easily swap out time period and location with some trimming of the skill list (eg probably no guns skill in the late Ottoman empire).
• GUMSHOE systems have more concrete rules (and thus complexity) than CoC, and also primarily focus on investigation (Trail of Cthulhu for CoC, Night’s Black Agents for modern-day vampire hunters, etc).
• Traveller (Mongoose 2E) is about travelling between planets in space, with a focus on trade, which can go from background noise behind regular adventures to “spreadsheets in space”.
• Worlds Without Number/Stars Without Number are generic, rules light fantasy and sci-fi systems that focus heavily on providing the GM with material with which to make a sandbox setting for the players to get involved with.
• Lancer is the communist tactical mech combat game.

For properties you specifically listed off:

• There are a bunch of Star Wars systems, from WEG d6 (very rules-light) to Saga d20 (basically D&D) to FFG’s three systems (decently complex with a focus on narrative, uses weird dice).
• Star Trek Adventures by Modiphius exists, but I’ve never played so no comment on quality.
• There’s an ASOIAF system out there where players play as one noble house, with things like hard-coded rules for social manipulation.

get the beginner box for pathfinder 2e! i strongly recommend it, best ttrpg product ive ever fucking bought, EVER. and ive played a lot of systems. the rub of it is its basically like a board game, but for dnd, and its easy as fuck to bring to family gatherings and such and hand out characters for them to play. the materials are very high quality and it comes with pawns and maps and a prewritten adventure and is probably the best intro to a system ive ever played. 10/10 no notes

“you can be the bad guys in this setting… even though they’re technically the good guys now but with a dark aesthetic to them!,”

Emphasis on this, ig

Seriously, so many web novels, manga, etc. have this “Let’s follow the bad guys… who never do bad shit and are essentially the good guys but with cooler aesthetics.”

To be honest: I don’t like “villain protagonist” stories anyway but I don’t like “villain protagonist fights the snobbish good guy and is probably good themself anyway.”

Mork Borg is a black metal ttrpg with the dopest aesthetic of any I’ve seen. I’ve never played it but I bought the book just because it looks so cool.

I personally enjoy WOIN a lot. It has a nice approach to building a character where you can slot in levels and improve attributes with XP. Classes are open and mostly non-restrictive, and all the attributes are useful to every class.