Yes, that's right. But it's also a completely different sort of lag, you can't compare network lag to input lag. With net lag in a game like quake you had to deal with players not really being where they seemed, but at least the game still resonded to commands when you issued them. It didn't take 150ms for your character to start jumping after you hit the jump key. That's unplayable. Also we had mods for dealing with net lag, the "unlagged" mod that let you shoot players where you saw them, not where you really are. It worked well if you were playing with people who didn't cheat.
Incidentally, I remember something else about playing Quake in those days: I didn't need anticheat, and nor did any of my friends. We ran private servers and did our own matchmaking. We didn't need anticheat because we chose not to play with cheaters in the first place. If any of us were secretly cheating, it wasn't severe enough to be noticed and therefore wasn't a problem. Anticheat only becomes "necessary" when matchmaking is taken over by the company, you're made to play with strangers instead of friends, and when everybody is encouraged to take it suuper seriously with global ranking nonsense that encourages people to treat the game like a career, not like a game. I really hate this trend in modern games. I'm sure it drives "engagement" but it encourages players to develop unhealthy attitudes and obsessive behaviors.
Instead, only “authorized providers” are allowed in most cases, that charge you a lot of money for the privilege of having a dedicated host, if at all. Worse is when the game company themselves provide the servers without the ability to choose.
When we could host our own, at home or at school or in the work IT closet as a “storage controller”, we could control the experience. This is why Minecraft because so popular, mods yes, Twitch Personalities, probably also, but the ability to host your own world with friends and hold them accountable for blowing up your house at school the next day was why it became a phenomenon with kids.
And of course besides that, games with local multiplayer.
I avoid streaming for that reason, but it's not unthinkable to play shooters with high latency. For years in the late 90s we played Quake 1 with a 150ms ping and had a blast. If non-streamers were disallowed that might actually be a good platform for fair competition.