Screencasts can be disabled for various reasons:
1. the service is not available (missing plugin etc.)
2. screencasts are not allowed by the session mode
(lock screen etc.)
3. the UI is invoked in screenshot-only mode (portal)
Currently each of those conditions is handled in a different
code path, which means that later conditions can re-enable
the button.
There's also an inconsistency whether disabling the button
is done via visibility or reactivity, which still allows
toggling the hidden button via shortcuts (although a hidden
button means that screencasts aren't supported at all, so
nothing will be recorded in that case).
Address this by updating the button from a dedicated function.
Fixes: 671df28a50 ("screenshot: Only handle mode-switch shortcut when
supported")
Closes: https://gitlab.gnome.org/GNOME/gnome-shell/-/issues/7358
Part-of: <https://gitlab.gnome.org/GNOME/gnome-shell/-/merge_requests/3155>
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