So I can confirm the message that the "please restart" notification seems to act up if the system tray is enabled and I'll look into it, but generally that notification is a short term solution while snapd will have an update soon that improves how it handles upgrades so that people would never see that notification again. You can enable it experimentally with
sudo snap set core experimental.refresh-app-awareness=true
And snap should actively wait for you to stop using an application before updating it. I think it's currently being held in experimental while decisions are made regarding how to give user feedback on what's going on regarding notifications from the system about updates being available.
If you wanted to disable automatic updates entirely, it is doable but at the same time there's a point where I'd also just recommend the AppImage official release if that's the case. You'd need to install the snap without accepting the assertation that verifies it as originating from the Snap Store (which indirectly means it abides with the Snap Store policies), which would look something like
snap download joplin-desktop
sudo snap install joplin-desktop.snap --dangerous
Specifically, ignore the advice it gives about running
snap ack. Because of the dangerous flag, it would then never update since from your clients perspective, it isn't actually on the snap store.
Ultimately the lack of control here around snaps is a pretty common criticism of snaps in general, and while there's options like limiting how often updates can be done and putting up specific versions on the store (e.g, I could put 2.2 up alongside 2.3, both with the same package name and offering the choice to the user which they'd like), in general I'd say you probably want the AppImage or the Flatpak which is less aggressive with the updates.
With the sync-target upgrade for example, I'm personally happy the automatic updates existed because it ensures the rollout of 2.3 happened as fast as possible for as many users as possible, and if someone really did want to temporarily go back to 2.2 because they had other clients they couldn't upgrade, there's a
sudo snap revert joplin-desktop for that!
So while people who'd prefer updates never happened will probably always disagree with the philosophy snap essentially enforces, I do think it's nearly at a point where the experience would be frictionless, going forward, people might not even realise they've been updated at all! (Cue horror music or dance music depending on which side you'd prefer here)