It may also not be as complicated as you think.
For example, unless they are monitoring catchers for pitch-framing, etc, how many plays does it really entail? There are maybe 50 batted balls in a single game on average, which is all they need to focus on. (This is a guess on my part, but with an average of 66 plate appearances per game, 9 strikeouts per game and an average, 6 walks per game on average, and maybe about 1-2 HR/game that they don't need to make a call on, it's probably close.)
By watching a game that is already over and fast forwarding to the batted balls in play, it won't take a full 3 hours to watch a game.
They also may have access to camera angles we don't have access to when we watch a televised game.
No one has said the system is perfect. In fact, everyone admits it is not. But that doesn't mean it is useless, either. In fact, right now it is the best system in place and th incorporation of StatCast data for launch angles and exit velocity has probably enhanced it even more...