Star Trek didn't even come close to inventing this. "All in the Family," "The Mary Tyler Moore Show," and "Happy Days" beat "Star Trek" at this by at least a decade, probably more.
They had spinoffs while the original show was still running? But, were those 'Take one character and and make a new show around them' spinoffs? Those aren't quite the same thing as 'Take the exact same show in a new setting with a new group'.
I think it's a little unfair to
Star Trek to characterize its spinoffs that way. DS9 was hardly the exact same show as TNG, and VGR was not at all the same show as DS9.
But yes, all the shows we've cited did have their spinoffs running simultaneously with the flagship show for at least part of their runs -- and even if they did start with spinning off one character, they were often trying to do much the same kind of show/format. For instance,
Rhoda and
Phyllis were sitcoms about the romantic and professional lives of young single women just as much as
The Mary Tyler Moore Show was, and
The Jeffersons was built around its loudmouthed, bigoted, opinionated lead grouch and the extended family that challenged/balanced him just as much as
All in the Family was.
Also, the three Paul Henning sitcoms I mentioned were not spinoffs in the "take an existing character and build a show around him/her" sense; they all focused on distinct, original characters but simply shared a common universe.
Petticoat Junction introduced the town of Hooterville, which was later used as the setting of
Green Acres, and characters from PJ often made guest appearances on GA or
The Beverly Hillbillies.
TV networks have always thrived on imitation. If a show has a successful formula, networks want to copy it. That's usually the incentive behind spinoffs -- to try to repeat an earlier success. The characters may be different, but the desire to make the format as similar as possible is hardly a recent innovation.
Heck, this goes back as far as radio.
Fibber McGee and Molly, a beloved radio sitcom from 1935-59, spun off another classic show,
The Great Gildersleeve, which ran from 1941-57, as well as
Beulah from 1945-54, focusing on the McGees' maid. There was also a
Beulah TV series from 1950-52; both the radio and TV versions were the first sitcoms in their respective media with black lead actors. Fibber, Molly, and their supporting cast also appeared in several feature films, the first two of which were crossovers with Edgar Bergen and Charlie McCarthy.