[I may try to flesh this out into a full-fledged post, but for now the idea is only partially baked. If you see a hole in the argument, please poke at it! Also I wouldn't be very surprised if someone has made this point already, but I don't remember seeing such. ]

Dissolving the paradox of useful noise

A perfect bayesian doesn't need randomization.

Yet in practice, randomization seems to be quite useful.

How to resolve this seeming contradiction?

I think the key is that a perfect bayesian (Omega) is logically omniscient. Omega can always fully update on all of the information at hand. There's simply nothing to be gained by adding noise.

A bounded agent will have difficulty keeping up. As with Omega, human strategies are born from an optimization process. This works well to the extent that the optimization process is well-suited to the task at hand. To Omega, it will be obvious whether the optimization process is actually optimizing for the right thing. But to us humans, it is not so obvious. Think of how many plans fail after contact with reality! A failure of this kind may look like a carefully executed model which some obvious-in-retrospect confounders which were not accounted for. For a bounded agent, there appears to be an inherent difference in seeing the flaw once pointed out, and being able to notice the flaw in the first place.

If we are modeling our problem well, then we can beat randomness. That's why we have modeling abilities in the first place. But if we are simply wrong in a fundamental way that hasn't occurred to us, we will be worse than random. It is in such situations that randomization is in fact, helpful.

This is why the P vs BPP difference matters. P and BPP can solve the same problems equally well, from the logically omniscient perspective. But to a bounded agent, the difference does matter, and to the extent to which a more efficient BPP algorithm than the P algorithm is known, the bounded agent can win by using randomization. This is fully compatible with the fact that to Omega, P and BPP are equally powerful.

As Jaynes said:

It appears to be a quite general principle that, whenever there is a randomized way of doing something, then there is a nonrandomized way that delivers better performance but requires more thought.

There's no contradiction because requiring more thought is costly to a bounded agent.