If you have a nice random mix of positives and negatives you could even put them in specially-marked envelopes and essentially dare the contestants to see if they feel lucky. Or not. The random elimination would suck bigly, but I really don't hate it, and I wouldn't put it in the first 4 or 5 legs either.
Although generally I believe a team's lack of skill or ability should be the sole cause of their elimination, the fact is in a race, bad things can happen. Even traveling on your own you can lose a passport, miss a flight, get robbed, get sick, have a lost hotel reservation, etc. So if y