I am wondering how miners are behaving toward their own transactions. Do they send their own transactions to the mempool or can they process/confirm them directly? I am assuming that the miners are interested or involved in sending and receiving of the bitcoins as well.
I suspect that miners submit their own transactions to the network like other users, but that they may prioritize them in their own blocks beyond what their feerate would merit.
The blockspace is limited to 4,000,000 weight units. Generally, miners simply fill it with a selection of transactions that optimizes for highest total transaction fees collected. Including their own transactions displaces other transactions from their block. If the displaced transactions paid a higher feerate than their own transactions, the opportunity cost of not collecting those fees should be equal to what would have been enough to get their transaction into another miner's block. Nobody can predict when exactly a block will be found or by whom. By submitting the transactions to the open network at a competitive feerate, the miners can get either quicker confirmation times (because any block will include the transaction, not just their own), or signal lower priority than the transactions they would currently include in an optimal block.
I would not be surprised if some miners included their own transactions in their blocks regardless of the feerate, though. It may even make sense to attach a competitive fee even when miners include their own transactions, because the transaction fee can be accounted for as a business cost, which is a bit harder to do for an opportunity cost.