EDIT: Maybe I completely forgot how Teamspeak works. It seems like there is a global friends list, but I can't remember that it was a thing back in the day (10+ years ago).
And that is what Discord alternatives will have to solve - the ease of setting up a new Discord "server" by any old random user is hard to beat in terms of convenience. Matrix is the only real alternative on that front.
However, if you have an established community and have at least a little hosting knowledge among the staff, the moat is shallow to nonexistent, and it's just a matter of how much of a pain in the neck Discord decides to be.
Teamspeak, as far as I know, doesn't have a way to solve this.
That being said, after thinking about it, I actually have done what you're talking about before - just not on Discord. When I find someone, I simply add them on Steam, PSN, or whichever account the game uses.