Shoot pool

is "shooting pool" a kind of game in the USA?

Pool is the game and shooting is the action taken during the game.

From Merriam-Webster online:

4 a : to engage in (a sport or game or a portion of a game that involves shooting ) : PLAY b : to place or offer (a bet) on the result of casting dice : to use up by or as if by betting : EXHAUST

Growing up in Brooklyn, NY in the mid fifties to early sixties, I spent many many hours/days in pool rooms where we played the game of pool AKA pocket billiards.