History of "this show filmed in front of a live studio audience"?

No, I understand. One of the Gilligan’s Island movies were done without a laugh track. It was a bizarre experience.

I think the Brady Bunch movies ingeniously got around that by musical cues or puzzled looks from the ‘modern characters’

Although I Love Lucy was never actually broadcast live the early episodes were filmed exactly like a play was being performed, in real time, for the audience. This was very exhausting for the performs and Arnaz quickly realized that the audience would wait tru “intermissions” while the sets were changed and the actors changed costumes. Later one the show did experiment with location shooting and filmed inserts.

Most drama and light entertainment (in England, at any rate) was done that way for many years, and audiences accepted props that did not quite work the way they had been supposed to or people fluffing their lines and strange noises happening off. Filmed inserts gave the cast time to get over to the next set and change their costumes if needed. Shows like Bilko and I Love Lucy were filmed with three cameras simultaneously covering three different angles and the result was high quality (technically) television, but it was very expensive as you never used more than a third of what you shot.