PC-TECH But the system reports an error...Gnome shell missing not installed...
That's expected behaviour in Budgie and not new. Budgie is based on GNOME, but it isn't GNOME. Budgie has it's own shell and doesn't come with GNOME shell installed.
GNOME tweaks is designed for GNOME, not Budgie. That it kind of works in Budgie is rather by accident than by design.
At some point in time I expect Tweaks to stop working in Budgie, as Budgie is in process of getting more independent from it's GNOME base.
But for now you are just fine by following pomon's instructions regarding the desktop file to make Tweaks reappear in Budgie start menu.
Even better: if you got some spare minutes create an issue for this on the Solus dev tracker in order for this to get fixed for all Solus Budgie users:
https://github.com/getsolus/packages