If there’s any way to fix this without changing the ID, that would be greatly oreferable because, of course, it’s the thing that identifies the message. I wonder what other problems would pop up if the message ID changes every time it’s edited.

To be clear, the message-ID in an outgoing email is not the same as the message identifier that appears in the archive.

From Wikipedia:  "Message-IDs are required to have a specific format which is a subset of an email address and be globally unique. No two different messages must ever have the same Message-ID. If two messages have the same Message-ID, they are assumed to be the same and one version is discarded." https://en.wikipedia.org/wiki/Message-ID 

So the question is, do you want all outgoing messages that show the edit to be discarded on the receiving end? That's what the standard is...thankfully, many do not follow it.

The proposal could have some effect on message threading, though. It could be better to treat the edited message as a "reply" to the original. I'll leave such details to the expert.

Regards,
Bruce