However, our current system tends to reward those that get there first, not those who maximize their chances of being correct.

Depends on situation, and overall, your description of choosing between bias and success sounds like a false dilemma.

Your scenario assumes that 1) reading the clues you find out the location of the prize sooner than if you just waited until someone finds it, but 2) after you found out where the prize is you would have no time to get it physically, and meanwhile 3) you want to get the prize and simultaneously 4) you want to learn where the prize is as soon as possible.

I wonder how frequent such situations are (I don't think it applies to science), but nevertheless 3 and 4 are in a conflict given 1 and 2. Conflict of priorities is hardly something unexpected, and rational advice is straightforward: decide whether you want more 3 or 4, if 3, take one door and run, if 4, study the clues. Are you suggesting that decision about your priorities needs bias, or that prioritising 3 is bias? In either case, you would be using the word in a non-standard way. LW wiki defines bias as a specific, predictable error pattern in the human mind. The word error is important. Bias isn't a catch-all category which includes everything that retards knowledge.