And if family member x plays one of my games, I cannot play a game or else they’d get kicked.

If you are doing Steam Family Sharing, then the only limit is that you both cannot play the same game at the same time. You can play different games at the same time, even if they are both from your account.

I use the feature all the time, it is great. Convinced my SO to buy a few games already as they tried it from my account, liked it, then bought their own copy so we could play it together.

Two people can play the same family share game if they are both in offline mode when they launch the game.