[This is incorrect. For some reason the AppleScript dictionary for Messages in the Script Editor Library was dimmed out in my Big Sur installation. However, after removing it and adding it back it is there, and after logging in the Messages with my iCloud account, I could successfully send an iMessage from Keyboard Maestro, so this was a false alarm - there are some other oddities, but the basic functionality works].
Just a note that as described on Daring Fireball, Apple has rewritten Messages in Big Sur as a Catalyst application, which does not support AppleScript.
As such, the Send iMessage and Send SMS actions will no longer work in Big Sur unfortunately.
Speaking of Messages on Big Sur, does anyone have a Keyboard Maestro work-around for the shift-arrow text selection bug? (It only affects Messages, as far as I can see, but not being able to select text via the keyboard is maddening! Option-shift-left/right-arrow works as expected, but not Shift-arrow.)
I’ve tried the usual suspects in KM, but they seem to have no effect; presumably limited by the underlying issue.