Mildly Interesting

I don't know what you mean by more descriptive. The only meanings here that are necessary are "by chance" and "unexpected".

The word 'uninentional' would convey a similar meaning, it'd be fine, but there are situations out there where it doesn't work.

"Their meeting was entirely unintentional."

By whose intent? If we're speaking from the perspective of fate, then this is actually a bad choice.

"Their meeting was entirely inadvertent."

I'm... not sure what this would even mean. I would probably assume this was a 17-year-old who got their hands on a thesaurus.

"Their meeting was entirely by chance."

This obviously works. I think it lacks a bit of whimsy. Maybe it trades one whimsy for another. Accidental means "by chance," though, you'd only be trading the word for its definition.

but sometimes it becomes ambiguous, as in this case (1-b is, too, applicable),

1-b is not applicable. The outcome of this accident is a fortunate one, so a misfortunate interpretation would be incorrect. I mean, this is why it feels odd to you, is it not?

I can only assume that what happened is you read the sentence and thought it would end "They accidentally shit their pants," or something, and were surprised it didn't. That didn't happen to me. And even if it had, I could have recovered.

I'm also on your side regarding "literally" [...] I don't use it as an intensifier. It's stupid.

Eh heh~ This is not the side that I'm on.

Knowing different sides of a word, or in this case adding new ones, allows you to affect your speech in different ways to say different things about yourself. If you're being formal, then you wouldn't use the intensifier variant of literally, or at least you would be more precise about it, but if you wanted to seem more street, or maybe youthful, then literally might be a great choice. Think about the power this gives you when writing dialogue for characters, or when speaking to certain demographics.

In my view, 'literally' getting another definition only increases the number of toys in my toybox.

I did say earlier that I defend slang.