I thought I'd update this idea with a solution to the problem as stated and also a great reason why no one will ever implement proof of burn as described in the OP:

The goal is to produce a design equivalent to Bitcoin, with the same objective consensus mechanism and security guarantees.

The major problem as stated is that miners can just dump a bunch of blocks mined in secret onto the chain to orphan the best branch instantly (Finney attack) to recoup the supposedly "sunk cost" of burning coins.

There is a way to mitigate this problem, which is to make the chain a DAG and have it be totally ordered by maximum cumulative burn. That way attackers burns always end up being a sunk cost because the DAG has a total order and no branches are orphaned.

The problem that this creates is no the is no competition to mine blocks, because every burn in the DAG counts, so every miner wins. This is a critical flaw.

The logical conclusion is to use the total order to define a rolling window of N burns, the largest of which will win the block reward. Once the reward has been won by a burn, that burn is no longer eligible to receive new rewards, the miner must burn again to be eligible. Of course, then it is optimal for every miner to submit N-1 tiny burns and one slightly larger one such that they always win the reward, leading to spam and chaos. To mitigate the problem you have to limit the block reward to the sum of the burns within the window.

Once you've done that, there is no incentive to mine because a miner can never win. If you increase the reward to be slightly larger than the sum of burns, then the coin will hyper inflate instantly because there is nothing limiting the time between blocks.

So, after all that we've learned two things, the under the heading Satoishi didn't make mistakes:

1) Without some objective measure limiting the time between blocks, you cannot have a block reward.
2) Without a mining incentive there will be no security in the chain

Hope you enjoyed this rambling and please feel free to shoot me down.

Cheers, Paul.