It's not a real practical problem – and I'm not sure why we're pretending that it is one. Let's clearly state what the problem is: OP wants to play MMOs & online games against potential threat actors who may DDoS and/or doxx them. Which is why they simultaneously also want to shield their identity from them. An analogy would be somebody who wants to play games with the neighbourhood meth gang while not wanting to get stabbed.
This is not safe or reasonable behaviour. Even if these people lack a client IP, a motivated attacker can piece your identity together from what you say and/or post eventually. Discord just makes that a bit more difficult at the cost of you handing over PIID of greater importance.
Most MMOs & online games offer an in-game chat client that does appropriate cloaking & is also moderated (so an offender risks a ban if they engage in harassment). Nobody is forcing OP to invite these people onto his VoIP server.