I mean, you can always retcon in a plainly after-the-fact sci fi explanation if you want.
The center of the galaxy can’t be reached except via a unique, super-rare yet scientifically predictable alignment of wormholes briefly forming along a certain route at the same time — hey look, one will actually pop up next year, let’s schedule an exploratory flight to the Center and back while we can!
And 20ish years later, a mad mutated Cytherian exile that thinks it’s God, sensing whispers of galactic-range Vulcan telepathy (CANON!), distantly and subtly manipulates one particular pliable Vulcan into sensing another such alignment and bringing a ship across it to escape its prison. Whose course, an ambitious Klingon captain rolls the dice and follows.
In both cases, the traversible route doesn’t last longer than a week, probably less. And because “the center of the galaxy” is actually a really, really large expanse of space, it contains any number of entirely separate weird locations and environments; the entry to Lucifer’s universe is nowhere near Sha Ka Ree.
Outside of these rare instances, you’re stuck, and crossing the galaxy takes seventy years or more.
Is this bullshit? Of course it is. So what? So is eighty percent of Star Trek, and that’s fine.