semper The flash of desktop followed by lock screen is normal for budgie-screensaver and its ancestor gnome-screensaver. It isn't ideal, but it gets the job done well enough for now until we either rewrite that aspect of the fork or implement our own solution baked into the WM for Budgie 11. Mixing GNOME Shell + GDM and Budgie is going to actually make locking even buggier, since then you have GNOME Shell creating a connection to the same DBus interface used by budgie-screensaver for locking, and it blocks budgie-screensaver from working. So yes, they are fighting as a result of changes in GNOME 40.
My recommendation is to either stick with GNOME Shell + GDM and no Budgie, or LightDM + slick-greeter (the GTK greeter on top of LightDM for login) + Budgie (including budgie-screensaver).
You could remove the entire desktop.budgie component to remove Budgie, however for actual installation of Budgie I would absolutely not recommend the entire component, but rather installing budgie-desktop-branding-fortitude-plus
which pulls in various defaults, Budgie Desktop, deps, etc.
Hope that offers some clarification.