DUELIST

This should primarily be done by those who have little time, but here it turns out to be the opposite.

There is no such priority order. And it is not written anywhere in the rules. And it is not even implied.
In reality, the first one to write is always the one who sees it earlier and knows when they can play.

The only difference is in who is writing. Half of the players cannot formulate anything more coherent than "hello" or "hello, I can play right now." This half is in a good position when they are online and playing non-stop. As soon as they go offline, it becomes almost impossible to agree on a time. The other half of the players can plan their time a week or two in advance, and it is possible to agree with them even when both opponents are almost offline.
And, in my experience, the proportion of players in these categories is around 50/50. That is, introducing any additional disciplinary rules will not be a popular idea, as it will be a burden for half of the community.

In any case, in order to write first and suggest timeframes, the opponent does not need to be online.