Because it was the first space exploration goal that President Kennedy was assured the U.S. could beat the Soviets in achieving.
The question we would have to reiterate is what would be the purpose? The moon landings aren't really a working analogy, for all the good they did they were about publicly beating a rival, sending a crew who would return and visibly achieve something the rival nation had not. Sending such a generational ship would be more of a blind throw of the dice. None would know for generations what had come of such a ship, no rival shows any inclination to compete in terms of putting a crew in another galaxy. They would just fly off into the unknown to effectively never be seen again. Giving them means of communicating or travelling at higher speeds would partially mitigate that issue but also take us back to square one with the show effectively becoming TNG with new terminology and bigger numbers. If a show were to revolve around competing with a rival power over some goal or other it would be far more feasible to have that goal be something closer to home, something that could be used as a strategic or propaganda victory. "Tin Man" or "The Chase" might be seen as archetypes for the way such a story might play out, as might DS9 having the Federation take control of the wormhole or provide a presence in the Bajoran system.
There are about hundred billion star systems in the Milky Way. If the Federation could explore one system in a day (and they probably can't) it still would take them over 250 million years to explore them all. I think there is plenty to explore in our galaxy...
'Cause it's next. 'Cause we came out of the cave, and we looked over the hill and we saw fire; and we crossed the ocean and we pioneered the west, and we took to the sky. The history of man is hung on a timeline of exploration and this is what's next. --- S. Seaborn
Spore drive wouldn't work, as you need the star system mapped in order to use it. At a minimum, the system has to have had a probe sent to it, mapped, and data sent back before one could use a spore drive to insta-warp to it.
Stranded in the Alpha/Beta quadrant, a thousand years ago, ten thousand years ago. Visiting familiar worlds, looking for the means to return home without violating the temporal prime directive. That could work, I suppose. I'd watch it.
Any story possibilities to be found outside the galaxy can be found within it. It's less setting and more production policy, simply don't endlessly write stories about the same short list of species (Klingon, Romulan, etc.) and introduce new species with new characteristics.
It did, but then it went back to the point that the sabbatical started, so we didn't even know it happened.
I think if they used the original premise of TNG and the Enterprise-D, and you had one single ship off in some galaxy without any backup, then I think that would be cool.
I think there is plenty to explore in our own galaxy. I mean, Star Trek already tried the "lone ship without any help in a strange area" and, love it or hate it, we got Voyager. Maybe it would be something that would work as a mini-arc in a series, I feel like there's plenty of themes, ideas and unknowns to explore here.
There was the idea that one might find a stable wormhole that leads to another galaxy. There would, however, have to be something quite different about this strange galaxy. Perhaps AI came to dominate this galaxy, and are divided into factions? AI might be quite strange compared to forehead-aliens-of-the-week.
1) Why does this have to be another galaxy? It could easily be in the still vastly unexplored regions of this one. 2) Depends on the writer. AI can come off as just as contrived and clichéd as Forehead Aliens. 3) Budget may require Forehead Aliens to pop up anyway.
Indeed, Wormhole, I agree. Nobody seems to come up with a convincing reason to have a show in another galaxy.
Because there really isn't any. Our own galaxy is enormous, and even in the late 24th century a large percentage of the Alpha Quadrant alone is still unexplored and uncharted.