Some poking around on TVtropes has lead me to discover that what you’ve previously called an “umbrella series” is called a Wheel Series on that site: tvtropes(.)org/pmwiki/pmwiki(.)php/Main/WheelProgram
Yes, exactly [link to page URL in ask]. Though the article doesn't mention Maverick or several other examples I like to give when I'm talking about it.
I usually bring up these types of shows when I'm talking about what I'd want to do if I were showrunning a Doctor Who reboot - not just a revival like in 2005, an actual reboot - which would have a pilot movie featuring origin stories for the Doctor's first three incarnations, and then the series proper would rotate between episodes with different incarnations. If the TVTropes editors don't regard Maverick or Search or Young Indy to be wheel programs, they wouldn't include my Doctor Who either.
