To get back on topic regarding burner addresses:
I tend to agree but the thing is nodes don't have to store OP_RETURN transactions. Those are provably unspendable so if you want to have something that is stored by every node forever then you have to store it in the utxo set.
Nodes do store OP_RETURN transactions, just as they store every transaction. What they don't do is add these unspendable outputs to their UTXO set. Any data stored using OP_RETURN outputs is still very much forever part of the blockchain.
you don't even know if the bitcoineater hash corresponds to a public key or not. there may not even exist a public key that hashes to it.
With (almost) 2
256 possible private keys and 2
160 possible P2PKH addresses, then it is highly likely that there is a valid private key. But yes, impossible to prove without finding such a key.