I've actually run into this helping a friend host a game server on their residential internet in a more rural part of Texas. They had to call their ISP and request a static IP address at an extra cost of something like $5/mo.
You’d just use the VPS as a proxy with a publicly accessible IP, and tunnel the connection back to your home. That would add latency that might be undesirable for a game, but maybe not.
Unless you're playing something like a turn based game, adding latency/hops is almost never desirable. Every multi-player game that I play is latency sensitive (and I don't play FPS games.)
