You don't need traffic shaping managed switches. You just need to traffic shape at the firewall. I assume you are talking about using 802.1Q QoS or DSCP traffic shaping? I can imagine it would be a bit tedious to setup especially if you are going to do it on a per game basis. If you had all Cisco , you would need to create several items , class maps , policy groups , ACL's to get it working.
Since most games use UDP for the actual gaming client , you could probably throttle HTTP / TCP connections at the switch level but that would take a pretty decent switch , which is above the budget of most LAN's .
Discord has a port set as well so you can make rules to shape it.
I am the LAN!!!!