You'll end up spinning your wheels on endless mistakes like this. I sat on my couch for months finding flaws in all my thought experiments.
I've indeed found a flaw in my assumption that an attacker would need 51% of the hash rate and 51% of the total burnt coins (or any equivalent combination of the two) to take over control. With the proposed design, you have to outnumber the creators of the last N blocks in order to be elected more than 50% of the times to build the second chain. To achieve that, you must possess more mining power (as a function of hash rate and burnt amount) than the lower 50% of the block creators, which is obviously less than 50% of the total mining power.
However, I think that the problem can be easily solved by using an alternating election rule for the minter of the second chain: For odd blocks we can use the described method, whereas the minters of even blocks are elected in proportion to the number of blocks they have built within the last N last blocks.
With such a rule, you could prevent that an attacker with only 51% of the sybils or only 51% of the mining power could control the chain.
You are assuming that the incentive for mining blocks on the first chain outweighs the incentive of Sybil attacking the accounts used to select who will be able to create blocks with transactions.
But there is nothing stopping any one from doing both.
Agreed. But the question is what costs and risks an attacker would incur by trying to take over the currency, and for what benefit.
In contrast to PoS, you cannot just buy and short the required stake since you actually have to burn it (or alternatively have a hash rate that is orders-of-magnitude higher than that of the rest).
To the extent you try to make the Sybil attack not free, you again reintroduce the economies-of-scale of resources and fall right back into the winner-take-all power vacuum.
If implemented correctly, a rational miners wouldn't get all the power over the transactions even if he had >50% of the mining power since the only way to achieve that is highly unprofitable. Of course, an irrational attacker with more than 50% of the mining power could attack the coin.
It's important to differentiate between the centralization of control of the global ledger consensus (which is tackled by my design) and how the consensus can be attacked.
These are two different pair of shoes.