Back in my day we had a thing called "server software" where we could host our own servers and there would usually be some active referee in the form of an admin on or available to call upon to take care of problem players instead of relying on a vote system that can be abused by the very problem players it aims to handle.
I'll wager a guess and say they're talking about Team Fortress 2, a game with a huge amount of community servers and most of the cosmetics being hats for your characters.
Most people who complained about that either never tried to find good servers, of were asshats that either didn't realize their the issue, or knew they were and are just trying to smear ppl caus3 fragile ego.
Doesn't this not really fix the problem, though? If the problem is that players are kicking others repeatedly for unjustifiable reasons, having a dedicated server basically just means the server owner retains the right to kick for unjustifiable reasons, the same way a host does now.
Being able to selfhost game servers and allow only friends to join is sweet, I wish more games still allowed LAN connections to a selfhosted server without going through online services.
It's not like there aren't other P2P live service games (can't think of any with dedicated server hosting but there's no reason it couldn't work the same). They verify your content with 1 server the devs host, and then it scoots you off to the actual game server which could be hosted by anyone.
Edit: Actually isn't that how Minecraft Realms work, kinda? You don't physically host it, but you can do whatever you want with it and it still works with the store content and such. I've never messed with one, personally so I'm not 100% sure. Only the Java version server.
Of course, those could also be modded and have more ways around verifying legit content you paid for allowing you to have everything for free, and that's what they don't want.
No idea how brink works. I actually just bought Helldiver's and played for an hour. All players are fighting against 2 alien races in shared missions. Start a mission by yourself and randoms can join midway. Each win pushes aliens back slightly and losses presumably push humans back. They also have a live game master that can manipulate missions in real time. Massive tuning would need to be done to run a private server of this.
Brink's "story" is told by actually winning matches as the side whose story you wish to see by having the missions come at you kinda like a push map, where whoever won the previous map advanced forward for their side, pushing the other side back. It actually kinda sucked if you wanted to know the story (not that it had a good one anyway) since there was a good chance you'd never actually win the last few maps since they were harder for the attacking team than the defenders.
From your description it sounds similar but on a bigger scale (and not PvP). I've still got a couple days before I can buy it.