Dang, wouldn't you run into serious internet problems with that? Do modern games use that much data? Could a single internet service handle 6 simultaneous connections?

Really is not that much data involved (not counting downloading the game itself). Streaming multi-player game data to/from an individual player can range from like 10kbps to 100kbps depending on the game and scale involved. Remember you are only sending the deltas (and occasional entire refreshes).

Latency is a lot more important than throughput for online games

Absolutely - especially when we are talking about first person shooters.

Lots of competitive games still have LAN options because they're basically required for reliable in-person tournaments. I'm familiar mostly with Nintendo games like Smash, Splatoon 3 and Mario Kart but even very modern games like Valorant have the technical ability to run LAN servers (even if the game's publishers keep the actual LAN servers closely guarded and only use them for large in-person events). And of course CS and LoL and similar have LAN modes