I don't think those first 2 sentences are true. Is it true for some fans? Sure. But, I think, overall, most fans just want the calls to be right so the game is decided by the players, and not the refs/umps/whatever.
And it's not about benefitting certain teams at the expense of others over years, or even multiple games. It's about benefitting one team over another in each individual game. "The NFL wants the Chiefs to win so they get more calls" type arguments come from a very small, but often very loud, segment of the population. Arguing against those people is pointless. There's no argument that can be made that missed calls are balanced between the teams in each game, though. That simply isn't true. 1 team always gets the benefit in each particular game. Whether from bad calls, missed calls, or a combination of both. Some of us just want to take as much of that out of the equation as possible. And often time the advantage is very small, and not even worth really discussing.
You'll never find me claiming the refs/umps/whoever cost a team a game. But pretending that one team doesn't benefit more than another in any given game is ignoring reality. And when you get into the playoffs that disparity is naturally looked at more closely. Especially in a sport like football where 1 game not going your way ends your season. Or game 3/5/7 in a series where you know 1 team is going home at the end of the game.
I don't think I agree with your layout of the replay timeline in sports, either. The leagues expand replay after it does work, not after it doesn't. I can't think of 1 league that implemented it, watched it fail, and said "the answer is more replay!" The NFL tried to replay certain penalty calls and stopped after it clearly didn't work. MLB started with just fair/foul replay, found success, and expanded to calls on the bases, etc. The leagues keep expanding replay because they are getting more calls right. Which seems like a pretty logical process to me.