at least one other word ending in “~begone”, but I can’t capture it. I have also tried to find other words derived from “bego”

Hmmm, are you sure you're not thinking of bygone, David? Was the first thing that came to mind for me. I've never heard of anything bygone except for times or days, come to think of it. Why wouldn't you talk of a bygone bus?

You also get begotten, used mainly (only?) in Biblical phrases. But that's another red herring (on a red bike) isn't it?

Water-muddying Fish