As far as I know, you have to Install gnome-online-accounts-gtk on Budgie.
And with that, GNOME Control Center isn't needed.
https://github.com/xapp-project/gnome-online-accounts-gtk
Edit: There was a brief period of time where Budgie users had to install gnome control center to set up online accounts. This got resolved with introduction of gnome-online-accounts-gtk.
Sync notes introducing gnome-online-accounts-gtk
Also making a first appearance is gnome-online-accounts-gtk, a standalone
application for setting up online accounts such as Google, Outlook, IMAP, etc., accounts
on GTK desktops. Now that GNOME Online Accounts has been ported to GTK4/libAdwaita,
it isn't usable in other control centers, such as Budgie Control Center, leaving users of these
desktops with no way to configure their accounts. The first application published under the
new XApp cross-desktop initiative, users of Budgie, XFCE, and MATE can again configure accounts
with this program.