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I don't understand overturning calls anyway. Aren't calls supposed to be umpires' judgments? What is the difference between overturning an umpire's ruling on a trapped ball call vs. overturning an umpire's ruling on a called strike, or safe or out at first base?
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4 eyes are better than 2, and 6 than 4 and so on.
I'll give you the most rudimentary example.
Imagine a close play with at first. The first basemen has to go up the line to catch the ball. The ball beats the runner.
F1V = Field Ump 1's View: It is his call. He will see the first baseman catching the ball w/ his foot on the base.
PUV = Home Plate Ump's View: He is watching the play and can see that the first baseman has pulled his foot.
The confer, HP overrules F1, the system worked.
Overturning calls is really, really necessary. There are bound to be time when an ump in proper position gets a bad view of a play. When there are more umps, they can confer and usually come up with the correct call.
If you have 4 trained eyes seeing the same thing, a conference of them all will usually illicit the correct result.
It's also important to note that, although there is an Umpire-in-Chief, these conferences are not formal.
Imagine the Manny trap (I haven't seen the play at all, this is only for example), this could be a conversation;
2BU; I saw a catch
1BU; Yeah, me too.
HPU; Nope, definitely trapped it
3BU; Yeah, doubt any of us could tell
2BU; Alright, we'll overturn it.
1BU; Okay, Jesus, this game is going really long.
2BU; Yeah, [censored] Pitcher is throwing a lot of balls
3BU; Yeah, hey, Rick, did you hit that young Asian girl last night?
etc, etc. They come to a consensus really quickly, in most cases, and keep talking to make it look more legitimate.