Good news, @ganncamp!
Yesterday I merged a core feature allowing a form to be submitted from a textarea with Ctrl/Cmd+Enter: this will work with any form that’s built using FormKit.
The tricky bit is that not every form in Discourse or in plugins is built using FormKit: your user note example is one such form. But, it’s a pretty quick job to switch to FormKit, so I’ve handled that in the user note plugin. 
I definitely agree with your sentiment that UI features that look similar should work similar, so here’s a small step towards that!