This is absurd: You thought up a detailed scenario, and you assign it ten times more probability than all other alternative scenarios of apologies put together:
- Harry apologizes in person
- Harry apologizes in writing (but without making a list of people versus Galleons)
- Harry apologizes through an intermediary (e.g. "Professor McGonaggal, please tell the Headmaster I'm sorry for the things I said", or "Fawkes, give Albus a hug for me.")
- Harry apologizes indirectly by defending Dumbledore's actions in public to Hermione's classmates.
- Harry apologizes through an elaborate prank that utilizes the Time Turner, the True Cloak of Invisibility, and exploding birthday cake.

Etc, etc...

One thing I learned from the Imperius debt experience is that this sort of impression can be misleading.

I thought that the possibility of Harry bringing up the Imperius debt was unlikely because it seemed like one of those crack fic ideas and the story so far was very much unlike those. What I failed to consider was that a lot of my impression of the plausibility of the story depends on the external details surrounding it: the writing style, Harry's internal thoughts, the reactions of the other characters, and so on. It was somehow a lot easier for me to come up with such details for other ideas, and so I thought they were much more likely.

Similarly, here, the "Send Dumbledore a conversion table" idea is already a Harry-like thing to do as stated, so it's easy to imagine in a story (it seems a little OOC, but not significantly so). "Just say he's sorry" is not a Harry-like thing to do at all, it's a very generic thing to do. So you don't imagine it being in the story.

But it's easy to wrap the plot feature of Harry just saying he's sorry in believable detail. Have him approach the table while angsting about some psychological experiment related to apologizing. Maybe have him start talking like a book for a minute and then stop and say "So, I guess, what I'm trying to say is... I'm sorry." Have Dumbledore put on a penitent, yet wise and understanding expression and say a Dumbledore thing. Now it's much easier to imagine, isn't it?

Edit: Or let's go the other way. You are, in particular, saying that Harry is ten times as likely to apologize to Dumbledore in writing as he is in person. Stated that way, does it strike you as plausible?

Harry knows he was wrong. His commitment to rationality and honesty might be enough to make him apologize. Most likely way he'll apologize, IMHO, is just tell Dumbledore what he was thinking, and why he realizes Dumbledore was right. However, the "Human beings can't live like that" line suggests Harry's inner conflict might keep him from apologizing.