The scope did shrink, but not in the way you suggest. The intent of the show's developers was that the E-D would be off exploring unknown space, far from the Federation and far from the old familiar Trek races, since Roddenberry wanted to make a new start (he had to be convinced to include a Klingon). The reason the ship had families aboard was because it was meant to spend as much as 15 years away from a familiar port, and few would be willing to commit that much of their lives away from their families, so the ship had to be a full, self-sustaining community, almost a mobile space habitat, rather than just a workplace. The show started at "Farpoint," the most distant Federation port in the galaxy, and the ship was supposed to just keep going farther out from there, literally going where no human had gone before.
But that quickly changed, since in the very second episode they were already coming to the aid of a Starfleet vessel in distress, and in the third they were on a medical relief mission to a Federation colony, and in the fourth (in production order) they were hosting Deanna's family and husband-to-be and hanging around a Federation protectorate, and in the fifth they were visited by Starfleet engineers for an upgrade. So the premise of going out into the unknown and staying there, the type of mission that the ship was specifically designed for, was largely forgotten immediately following the pilot. And it only got worse as the series continued and the focus shifted more and more toward interstellar politics and relief or rescue mission to Federation worlds and visits by or to the crew's families.
But no, they weren't planning to leave the galaxy, not with so much of our own left to explore. It wouldn't have made any real difference to the storytelling anyway whether a remote alien empire was in another galaxy or just a different quadrant of our own.
Besides, the reason "Where No One Has Gone Before" involved the ship leaving the galaxy is because it was loosely based on Diane Duane's 1983 TOS novel The Wounded Sky
, in which Kirk's Enterprise
tested a revolutionary new stardrive that sent them into ever deeper intergalactic space but damaged the fabric of reality, eventually creating a rift to another universe. So it didn't reflect any master plan on the part of TNG's producers; it reflected the plot of a novel written years before TNG was conceived.
It's called a Galaxy-class ship because it's intended to explore a galaxy, not multiple galaxies. If you wanted to build a ship to explore multiple galaxies, you'd have to name it Local Group-class or Universe-class!
Vonda N. McIntyre would dispute your reasoning. In her 1982 Wrath of Khan
novelization, she posited the existence of a new type of starship called the Galaxy
class, which was designed for intergalactic travel.