re: Which sport gives the least amount of control of the outcome to officials?

Football - refs can call penalties to move teams up and down the field and get them in scoring position

Basketball - refs can call ticky tack falls to get players to the FT line for free points

Baseball - inconsistent strike zone can get baserunners aboard

Hockey - send a guy to the penalty box for a couple minutes, but the kicker is the team at the disadvantage on players can now ice the puck without repercussions to waste time.

Hockey is definitely the sport (at least in the major 4 NA leagues) where try as they might the refs have the least influence. And the NHL has the best postseason of the 4 major sports as well.