I think with a single ship, you can see quite a bit of the Federation. The last three Trek shows, however, either were confined to one area or were somewhere the Federation didn't exist. TOS and TNG had a mix of stories both within and outside the Federation--both the original Enterprise
and the Enterprise
-D were really more multipurpose vessels, with exploration not being the only reason they were out there.
My own view regarding continuity is that it should only be treated in a general sense. The more you try to "fix it" or shoehorn it to fit the real world, the more problems it creates. Leave certain details vague and don't affix dates to every single thing that happens.
But I also think a new Trek series should focus on the characters and the setting more than "alien of the week" stories. The last two Trek shows had characters that were simply there or faded into the background because the writing staff focused on the easier-to-write-for characters or the characters they personally liked the most. In that sense, a more diverse writing staff led by a showrunner who wants more diverse stories is definitely required, IMO.