Board Game Arena

Would anyone be in favor of an option that picks a random game from "My games" list to be played?

Not me, but there seems to be a fair number on Yucata who like doing that, so I see no reason a fair number of people around wouldn't maybe also enjoy that.

https://www.random.org/

while you're waiting.

Simple lobby is close to this. You can choose to start multiple games and whichever fills first will start. Not random, but as close as you can with how BGA is currently set up.