I usually have one toolbar with buttons assigned hotkey sit on edge of screen right. I move these buttons on toolbar to lister window and delete this toolbar after clean installation of Windows 11 22H2. And now hotkey is not working if no lister open.
Is this normal?
Yes, lister toolbar hotkeys are not system-wide. If they were it would break the hotkeys in every other program.
Make system-wide hotkeys if those are what you need.
How can I make button in toolbar with system-wide hotkey?
In Customize has system-wide checkbox but only in Hotkeys group. No system-wide checkbox in Toolbars group.
You can't. If the button is in a lister toolbar and you want the same action to have a system-wide hotkey, then you need to make a separate hotkey as well.