Can you articulate where in the DMA where it says that all Apple has to do "is promise not to sue anyone" to be in compliance. Or where it talks about protocol publication.
The DMA is about preventing gatekeepers from using arbitrary restrictions to prevent competition. It does not try to predict and anticipate every possible example nor solution.
The existence of competitor would by itself be enough proof to the fact that Apple is not restricting competition. But for such a competitor to exist, they would require enough assurance that the business will be viable and they won't get sued out of existence.
Apple either publishing the protocol or at the very least publishing an official licensing agreement allowing anyone to reverse-engineer and reimplement said protocol would achieve this.
Evidence could be that Microsoft isn't in trouble for shipping Windows with RDP servers/clients in Europe, which is equivalent to this iPhone mirroring feature.
Why is Microsoft able to do it just fine (without running afoul of the DMA) while Apple supposedly can't, despite MS having an ever larger marketshare of its field than Apple and this would warrant even more scrutiny?
The multitude of third-party RDP clients (and nobody being threatened with legal action for implementing one) out there may be at least part of the answer.
Hint: it doesn't.