Though not named in the episode, the band accompanying Gomer is indeed the actual Marine band. The closing credits state: "Special guest appearance by The United States Marine Band."
Producer Dan Merrill is likely based on real-life Broadway Producer David Merrick, who had a string of hits in the 1960s, such as Gypsy, Oliver!, and Hello Dolly!
The title comes from the common idiom that no matter what the problems with the performers and backstage personnel, the show is more important because there is "No people like show people."