qsl That's specifically with no other applications running or docked.
I'm not quite sure what you are talking about here, but the taskbar can be set to bottom, top, left, right,icon position on the taskbar is flexible, and the taskbar can be set to autohide or not when applications are open. Budgie is reasonably flexible, although not infinitely so.
qsl Or I could be horrendously mistaken and should give Budgie another chance.
Mistaken or not, why? You seem satisfied with Plasma. If you like Plasma, it probably makes sense to keep using it.
qsl In either case, not being able to arrange icons would also be a no go for me (never got far enough in the process to discover that).
The icons, both pinned apps (in my screens, left on the taskbar) and applets (in my screens, right on the taskbar) can be added, deleted, and arranged.
qsl It was like last month. At this point, just assume I am an unreliable narrator or maybe prone to hallucinations 😅
Were you using Solus Budgie? If so, I'm puzzled because Solus Budgie has been using the layout shown on the download page for several years. Other Budgie implementations use different arrangements. Ubuntu Budgie, for example, defaults to dual top/bottom bars, with center app icons on the bottom bar, unless you change to a different desktop layout in the Welcome Center, as I did.