Combined with your previous statement, that means that adults don't want teenagers to vote because they would vote for other rights, but not out of fear they would actually get them. Which is decidedly odd.

I've thought about it, and I think this is more correct, and I was wrong before. (Or perhaps both reasons are correct, but this one is much stronger).

Historically, minorities who had voting rights but were otherwise legally discriminated against - blacks, gays, etc. - didn't abolish that discrimination by simply voting themselves more rights. They had to fight for those rights by much more direct means.

Acquiring the vote is usually, historically, a relatively early step in the enfranchisement of a minority, and it doesn't help directly in acquiring other rights. (I may be wrong about this; I'm not an expert.) When adults imagine the scenario of teenagers gaining the vote, it pattern-matches the narrative of an oppressed minority more or less violently fighting to gain other legal rights. Adults don't want to give teenagers the vote because it would acknowledge them as an adversary, an independent force. It would give them some (perhaps symbolic) power and simultaneously frame them as opponents.