While I think such a series would basically be more or the same or any concept could be done in the Milky Way, there is something about another galaxy I find appealing for a setting.
While you could use a wormhole or even I suppose the gaps between spiral arms to give the crew distance from the Federation, those are not insurmountable problems, just time consuming. A starship in the Milky Way could still get home eventually as shown in Voyager and have any number of places to visit on the way there.
However if you set the crew in another galaxy with only a specific route back to the Milky Way, there is no other way to get home effectively. One cannot really go the long way to Earth without it being a sleeper ship or a generational ship, as the distances are too great and there is more of less nothing between the galaxies to keep the series jumping from planet or the week to planet of the week. So the story can't be about them trying to get home.
The series would have to be about settling a region of space with their only being a long support line back to the Federation, be it a warp gate, a wormhole, and hyperspace conduit, or some such thing where there is effectively only one way to get between the galaxies within anything less than centuries, and even that route takes many years to cross. Thus even if the ship can make it back to wherever, it will still take years to go home, so they are effectively on their own for seasons at a time.
They might have to defend their supply line from native species, if there is only a fixed point of entry into the galaxy, or if they can use the tech anywhere along the Great Barrier, they would have a specific base of operations and a homing beacon for other Federation ships coming and going on their say seven year voyage between galaxies.
Most of the rest of the series would be stock Star Trek like it has been since the 1960s, just with there sometimes being a reminder that they can't go home quickly, and might have to put up a defense (military or diplomatic) for their base of operations, or their pocket section of the Federation in the other galaxy. Depending on if we are watching the first pioneers, or one of the early waves of colonization/exploration after the "beachhead" is setup and any exploration region of some hundred light years is established for them to poke around.