Update: Creating a new Quick Note from the bottom right hot corner seems to be a suitable workaround for now. I recommend trying this first before attempting what I mention below.
I've come across this bug on a number of occasions, but it typically doesn't last long enough to try troubleshooting it (typically I just open BBEdit and copy it over to Notes after it plays nice and lets me create a new note again).
During the most recent time this happened, I found a response on the Apple Developer Forums that this bug occurs when your iPhone is not nearby and unlocked.
Quitting Notes app on macOS, unlocking my iPhone with it near to the Mac, then opening Notes app on macOS appears to solve this issue when it came up.
It's unclear if this is a bug or an undocumented feature. I have only tested this once. If it appears that this solution is consistent, I will mark this as the answer.