I've basically done 2 things this thread - suggested that one idea of someone else wasn't too bad, and then reiterated my own idea.
The best suggestion, for mine, would be to have all Umpires and Referee wired-up, so that the second an appeal went up you could have "looked pretty good to me Tim, what'd'you think?". "Just 5 secs Rudi, let me check it didn't pitch outside leg"... ... ... ... ... "nope, that's fine, and it was obviously hitting, and it was nowhere near his bat". "Okeydokey" Up goes the finger.
You could even have:
Appeal goes up for caught behind, Rudi puts his finger up
"Wow, wow, hang on a sec Rudi, not sure that wasn't thigh-pad"
"Oh OK Tim, have another look for me"
"Yeah, that was thigh-pad Rudi"
Rudi puts his arms over his chest to reverse the decision.
Simple as. Quickest way to do things, no need for a minimum number of anything, and the players still have zero say in what happens.