On the other hand, you should be gaming over a wired connection, not with a Wi-Fi link in the middle.
No it wouldn't. It's a separate WiFi device, using a separate allocation of bandwidth. How would it have any effect?
It will affect the latency on the actual internet link. The second you add extra traffic latency shoots up...it doesn't matter where that traffic comes from. You can mitigate the effect somewhat with QoS but it doesn't go away.
Bufferbloat on the upstream pipe, if somebody is using your shared connection.