TLDR; Use poker decks to clone 3 sets of Love Letter

I'll recommend some boardgames.

Once you go over 7-8 players, the options are greatly reduced. The available genres become (mostly) limited to social deduction

Social deduction, all about trying to figure out other peoples cards. And lying, lots of lying.

• Love letter, 6 players, 20 cards, ~13 tokens, for points
• Avalon, 7-10 players, ~30cards, tracking mat/paper
• Secret Hitler, 7-10 players, ~30 cards, tracking mat/paper, has a print-at-home version.
• Ultimate werewolf/vampire/etc, ~10-12 players, ~13-15 cards, phone app (for narration, free)

*None of the above are very good for playing as teams (i.e. two ppl taking a single player spot)

Group Games games that do well with teams

• Codenames, various versions.
• Forbidden Island/Dessert/Sky - co-op game for various number of players

As a special mention, Challengers is a board game that allows you to create simple yet fun card drafting tournaments. There are two editions, each allows you up to 8 players, and if you have both they can be combined into 16ppl tournaments

For your numbers, you'd need to buy 2-3 copies of the same game. Not ideal, even if you have the budget.

I'd say, print yourself three copies of Love Letters, you can find the cards online, or just play with 2-3 poker decks. Its the best game in my library, simple yet deep. We've been playing it at least once a week for 3 years now. I've even managed to play once with a mob of disinterested teens ;)