I'd go mid to light games that are easy to learn & little setup:
Cascadia ($40)
Lost Cities ($20)
Cartographers ($20)
Draftosaurus ($20)
King of Tokyo ($30)
Kingdomino ($20)
Scout ($20)
6 Nimmt ($20)
Ticket to Ride ($40)
Carcassonne ($35)
Codenames ($15)
So $280 for an excellent collection.
Cascadia ($40)
Lost Cities ($20)
Cartographers ($20)
Draftosaurus ($20)
King of Tokyo ($30)
Kingdomino ($20)
Scout ($20)
6 Nimmt ($20)
Ticket to Ride ($40)
Carcassonne ($35)
Codenames ($15)
So $280 for an excellent collection.
Comments
I would keep Carcassonne, Lost Cities, Cascadia, and Scout. I would add Azul, 7 Wonders Duel, Heat, Camel Up, Decrypto, and one slightly weightier game like Castles of Burgundy. Total budget of about $350 or so?