I use multiple spaces and multiple screens on my Big Sur laptop.
For annoying reasons, I have to end up shutting down and restarting my laptop almost every day. (I have to use VPNs, and then a VMware Horizon Client as well, and if I don’t reboot, the lag inside my Horizon Client desktops makes it unusable.)
But whenever I restart Notes, it doesn’t come up in the screen where it was when I shutdown. In fact, if I start Notes, move it into the space where I want, quit it, and then restart, it will move right back to the other space.
Even if you don’t have/use Keyboard Maestro, there’s some information in that post regarding Desktop Spaces that you might find helpful. Also, you might know this already: you can assign applications to specific Desktop Spaces. I myself don’t assign applications to specific desktops, but I do assign the application Tot to All Desktops.
If you have any follow-up questions, I’ll do my best help.