Such transactions should not be broadcasted, because there's no single mining pool that monopolizes the network.
I never said they were. I said they could be a miner making use of block fees to avoid taint analysis, with the implication I was making that said miner does not broadcast the transaction but simply adds it to every candidate block they are working on until they successfully mine it.
And since it should stay locally, to the candidate block, there's no reason to not make a non-standard transaction. Such as one emptying the entire input in fees.
But such a transaction makes it completely obvious that is what the miner intended to do, whereas one which sends a small amount to another address and a huge amount in fees can be plausibly denied as "I fat fingered the fee" or "I was trying to code a transaction manually and forgot to specify a change address, and so everything was taken as a fee".