Why, precisely, is it irrational to respond to this?

Look. The basic assumption here is that people would rather not be the targets of a hysterical person's fits, that it's unpleasant to them, and if there were anything they could do to discourage it, they would do it. Another assumption is that people remember which strategy worked on a certain person when they tried to achieve a certain goal in their interaction with them. Therefore, if someone wants to get you to stop being angry at them, and they throw you a hissy fit, and it works, and when they tried to reasonably talk things through with you it didn't work, then they'll remember that the best way to get you to stop being angry at them is to throw you a hissy fit. The next time they'll want you to stop being angry at them, they'll probably throw you another. You don't want this. You're uncomfortable when they do that. Therefore the rational thing to do is not to reinforce that sort of behavior, and reinforce instead the behavior that makes you feel comfortable.

I didn't say anything about the rationality of responding to anger per se. I just said that reinforcing a behavior you don't want to be subject to is irrational (and I thought any audience could agree with me on that) and that this particular case belongs to that class of irrational things to do.

Did evolution prime us to respond to it because it thought it would be funny?

Why, yes. I earnestly believe that evolution has a sense of humour which influences its "decisions" regarding what sorts of behavioral tendencies to implement in humans.

It's disingenuous to suggest an answer to your question which you expect no reasonable person to give.

Maybe your logos isn't as good as you think it is?

Possibly, but I am not very tempted to fault the quality of my logos for the failure of my attempt at mediation, since the obstacle it had to overcome was of the kind "I don't want to listen to you. (I want to indulge in my anger.)". The only response that the other person would accept of me was to shut up, admit to not quite qualifying as a human being because of my moral faults, feel horrible about it and leave the room. I am inclined to believe that moderately unpersuasive arguments don't block the way towards eventual reconciliation quite as much as that kind of attitude.

You apparently weren't able to discern why they were upset with you in the first place, which certainly would have placed a damper on your ability to articulate convincing reasons why they should not.

I was very much able to discern what they were angry about. I just said I couldn't agree to their reasons, i.e. that I don't believe their anger was in the least bit justified. So, this ruled out the possibility of internalizing their accusations and feeling guilty.

It is, indeed, not obvious to me (though perhaps I have low IQ)

Stop this. Seriously.