This is the actual hard part of the marriage/relationship that the commenter higher-up is talking about. Compromise is table stakes. Communicating well with your partner, making sure they are an equal participant and making sure they feel heard and valued is what's important.
Again, focusing on who is going to "win" is not a healthy perspective in my opinion. Questions like "do feelings beat rationality?" Is the same thing as saying "does paper beat rock?" The purpose of the article was not that dishes and minor arguments about dishes on the counter resulted in a divorce. The point is that she felt belittled and unheard. He was winning the dish battles, but lost his marriage.
>Should feelings always trump?
No! But they might! Each marriage and relationship is different and they change over time (as people tend to do). For one couple this could be true, but that doesn't really mean anything. Again, this approach is thinking about marriage as a series of small arguments to win, with your spouse as the opponent.
Compromise is table stakes. Whether the dishes got done or not really doesn't have anything to do with whether or not these people got divorced, the issue was a lack of communication and one partner not feeling heard.
This is a less-bad framing of the earlier question, but it still suffers from the same flawed perspective. The goal is to have a healthy and happy marriage where both individuals feel heard and respected and like a team.
What if you feel more strongly about something than your partner? Do you feel like your thoughts and feelings on the matter are being heard? (Note: feeling heard is not the same as "I got my way") Do you feel like an equal team member in the decision? It's okay to feel annoyed or frustrated at the result, that's part of life, you don't always get your way. But it's a huge difference between "I felt like my opinion was fully heard and considered, even if we didn't decide on my point of view." and "I felt dismissed, belittled, and unheard. $spouse/boss/friend/person didn't really listen to what I had to say and I don't feel valued."