There's not meant to be any end to the powers they have on the field. Umpires aren't politicians or police officers, they are the be-all and end-all of all decisions on the field. The ICC can reprimend or even sack them after the match, but during the match the umpires have ultimate power. As such, the game can only function if both teams accept the umpire's decision on the day, and any umpire which isn't at very least impartial with any errors made will be held to account by the authorities in the game afterwards.
That's the way cricket works, and indeed the way pretty much every sport works. I can't imagine that you haven't realised that until now. And really, the umpires do everything on a hunch. LBW decisions are made on a hunch, for instance. You can't prove conclusively whether or not they are out, just like in most cases you can't prove conclusively that ball tampering has gone on. Even if you have a photograph of a fielder running his nails along the seam, it's pretty hard to differentiate between picking the seam and just cleaning it or whatever. The only thing the umpires can use, aside from witnessing strange behaviour with the ball, is what state the ball is in when they check it at the end of every over. If it's in a state which can't be explained by the usual way a ball would be treated, then they are well within their rights to assume ball tampering has gone on. It's happened before, after all.
The thing that really annoys me about the debates about this decision is the idea behind it all that the umpires are quite simply completely in the wrong to have punished Pakistan for tampering with the ball. Full stop, end of story, they shouldn't have done it. I mean, how dare they enforce the rules as written with reference to their views of what was occuring on the field? They surely should have just done absolutely nothing if they suspected ball tampering was going on, without a taped confession.
Really, what possible conclusive evidence could be brought forward which would make the decision right suddenly, in the minds of those who decided as soon as it happened that it was incorrect? Has there EVER been a ball tampering decision where everyone agreed on what had occured? There's always doubt, and the role of the umpire is to sort it out. Fair enough if Pakistan didn't think they'd tampered with the ball, but that's what umpires do, they make decisions that not everyone is likely to agree with. I don't think much of Hair and I won't shed a tear if he's dumped, and chances are no real ball tampering went on, but it's just like any other umpiring decision - you move on and you play the game, and complain afterwards if you think it was unfair.