the episode clearly shows that they have a skewed view of Voyager. And they needed the Doctor to correct them.
By the time the episode was over, they WERE corrected. They no longer had that skewed concept of Voyager
as a warship. That was the Doctor's doing.
Also, the Doctor would not need to start a long track home, he could just call the nearest outpost or ship and get a lift to Earth.
We don't know how long it would take for the Doctor to reach Federation space. Assuming this planet is not a member of the Federation, he might have had to do a bit of searching for the nearest Federation world. Still, that might not have been too far away.