If you unentangle the first example the author gives, she wants to give a task to the man, without explaining how to do it, and then complains that they don't do it properly. But it's not even just that, it's her wrapping this task into a "Mother's Day gift", i.e. forcing the man to do emotional labor to figure out what the woman really wants and how exactly to do it. This is a counter-example to her overall argument, both in her literal words(about emotional labor), and (I'm guessing here) in the underlying meaning of "men should do more of the hidden work"(how do you expect them to do the "hidden work" if you never explain to them that you're doing it, you want to share the responsibility, etc?).
If there's a deeper underlying message that I'm missing here, please, I'm interested in knowing about it. But every time I read an article like this, all I can think of is "dysfunctional relationship" and "unreasonable expectations".
This is where you're wrong. The wife makes it very clear to the husband how she wants the job done. This is clear because the husband reluctantly acquiesces to her request before deciding to go with a different approach. Once you accept that the wife's instructions were clear, the rest of your argument falls down.
Only if the delivery is offensive, otherwise the message is more important.
To give an example, I'm a native English speaker, and when I speak with some non-native English speakers they can make some quirky decisions on how they deliver their message. However, if I've understood the underlying message, I'll look past it, and address what they meant.
To bring it back to the topic at hand, meta discussions like the one we're having now often start out as a form of deflection. Regardless of the term 'emotional work' (I prefer 'hidden work', you may have a different preference), do you agree that the underlying message of the article is broadly correct? It's not so much about who takes on the 'hidden work', but rather it's about the toll it takes.