Does playing for the POST sport team affect career progressions and promotions?

I did in San Diego in 1970, and it got me recognition by base brass. That's a good thing, especially when we won.

Can go different ways. I played on the football team at Camp Lejeune which was a great time being a low ranking lance corporal at the time playing in a league with higher ranking officials kinda broke me in to how the marine corps works. Then turn around to playing Softball as a SSG at Fort Dix for 2 years had its ups and downs. Lots of crazy stories with that team.

Makes for a good bullet on your NCOER, but that is about it.

Generally you play on a team that has a cross section of ranks. That can be a good thing or a bad thing. Good in that demonstrating you're a good team player and respectful of the game, competitors, and officials. Name recognition gets around. Bad if you're the opposite.

I'm a soccer referee and used to do the MIL league out at Bangor. The Marine Barracks had a team. I booked one of their players and he demanded to know my name. I said "Referee, Sir" and asked if he wanted to continue to play. He grudgingly did. After the game I was changing my shoes by the car when the Marine and his Gunny walked by. They saw the screaming eagle on my windshield which evoked a strong slap to the back of the Marine's head from Gunny along with colorful commentary. Great to let the NCO community deal with their problem children.

Only if your sport involves tackling the General's daughter......

My best guess is it gets your name out there to some NCO's and Officers.