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Imagine every potential miner has a unique representation of identity, which they do (their address or public key). To get a concrete list for illustration purposes we could take the list of all miners who have ever minted a block in the past: now we have our list of miners.
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I'm merely trying to prove that Proof of Random is possible and, moreover, trivial. Can you see anything else I might be missing?
I guess this would trigger a rush by miners (or candidate ones) to get a new pubkey/privkey pair after each new block, to win the "randomness challenge". I means, I see the new block, so I start to check many privkey, choosing the one which produces the right pubkey, given the challenge metric. Which is again a sort of PoW.
If you avoid this permitting only pubkeys from the old miners list, you will prevent new miners entry damaging decentralization.
Just imho of course (the problem here seems not your idea of randomness, but to find a parameter -used to calculate the candidate value for the challenge- which can really be unique for each contender)