The snatch should engage all major muscle groups. You’re using your legs to get the weight off the floor, your shoulders, lats and arms continue pulling upwards, until the moment you must switch from pulling the bar up to pushing it up, again using the large muscles in the legs to help do the majority of the lifting. The tricky part is the transition from pulling to pushing because at that moment, you’re not exerting any force on the bar at all and it wants to drop, so you have to move faster than it. You accelerate the bar off the floor, pushing with your legs and pulling with your upper body as fast and as hard as you can to give the bar maximum momentum, and then when you’ve got it as high as you can, you drop under it and make the transition from pulling to pushing,
The simple reason you can’t lift a heavy weight like a broom is because it’s heavy and it requires a different technique.