You've said all that I was thinking. I even thought that they were probably trying to make this and open ending. But it still wouldn't make sense since there's a lot of plot holes.
I'm just gonna add a bit on the kids, part. She did mention her plans when her friend gave birth. She felt flustered when her friend asked about her family plan but she recovered her composure fast because to her, tea expansion within the empire is more important and she knows that the duke will accept her with or without kids. So we won't know if one of them is infertile or not especially since kids is not in her plans for now.
And I'm sure they probably have a contraceptive option, probably developed by magic. I don't know, I just presumed since the birth rate are not that high with her married friends or tea companions.
An open ending*
People saying that it is okay for them not to have kids, and that's fine...if it were a modern story, but this is set in a time and place where women still have very little rights and are considered to be inferior, the fact that her not having a child was never addressed or played a vital focus in the story is odd.
The issue, is that it was never addressed:
Is she infertile? Is he?
Are they waiting on kids, in order to have more time as a married couple?
Are they going to adopt?
Did they decide not to have kids? What about the Duchy succession? Does the Duke have a qualified relative that he is willing to pass on the title to should he pass away? Why didn't the king press them on this (since it is his duty to make sure the nobles fulfill theirs)?
When you take on a title, you have a responsibility to procure an heir, because what will happen to all the people the Duchy is responsible for and are under its care? They are not some peasants living a simple life, they decided to stay on as a Duke/Duchess and have all of the benefits that come with the title, but they also have duties that need to be fulfilled.
All of that was glossed over or never addressed. Again, not expecting much from a story that solved every issue by throwing tea at it, but come on..