I used to be in a draft similar to this when I played in Strat-O-Matic tournaments. They called it "Grand Prix" drafting (with "Chip Accumulation"). The general concept is the same, but there is no blacklist. Everybody would submit their player. If no matches, that person would get his choice. But when there was a match, we did something slightly different.
In round 1, if two people selected the same player, he would be awarded randomly to one of the two people who claimed him (coin flip, high card, etc). The person who won the player would have to give the losing person a chip. (If three people selected the same player, the two losing people would have to give a chip to the winning person).
In subsequent rounds, if two or more people selected the same player, the person with more accumulated chips would win the player and have to give a chip to the loser(s). Again, if two people with the same selected player had the same number of chips, the player would be randomly assigned and the winner gives a chip to the loser.
This methodology would, in effect, offset somebody getting extra lucky or extra unlucky on the random draws.
Something to think about.