

For me it’s GOG first. Using lgogdownloader and wine directly (in a custom apparmor profile). No DRM, no forced updates, no annoying client that takes forever to start. Games are also dramatically much easier to isolate and sandbox this way.
If the game is not there, then yes, Steam (as a separate unix user).
I’d go for HLS due to its simplicity: just files over http(s). VPN or not - depends on your network. If your machine is accessible from the internet, just putting the files into a webserver subdirectory with a long random path and using https will be secure enough for the usecase. Can be done with an ffmpeg oneliner.
The downside of HLS is the lag (practically – 10s or more, maybe 5 if you squeeze it hard). It is in no way realtime. Webrtc does it better (and other things too), but it is also a bigger pain to set up and forward.
Also, just in case, test that the webcam works fine if left active 24/7. I had (a cheapo) one that required a powercycle after a week or so…