Voyager was sent 70,000 light years away to the Delta Quadrant, because there wasn't anything left worth exploring within the 8,000 light year expanse of the Federation.
Star Trek is suffering from Power Creep something awful.
This is incorrect.
Voyager was set 70,000 light years away so that the things that they would be doing on that show would not cross over and create continuity issues with DS9 and, in particular, with the canon timeline that was expected to be in use in the restarted Star Trek movie franchise at the time. The "political" story of the Federation and its allies and enemies was something that was to be the subject of the existing movie series, and to a lesser extent, with DS9. The same reason was the excuse as to why DS9 was doing its thing through the wormhole. Again, the political situation in the "main" Federation space was reserved for TNG and the movie series. As such, Voyager was to return to pure exploration mode in the Delta quadrant in a manner which would not interfere with the continuity in existing programs.
OF course, we know how that ultimately worked out. We ended up with new political stories and issues that were particular to the Delta quadrant, which ultimately nack-contaminated Fed Space after DS9 ended and the movie series fizzled.
Star Trek always takes on a life of its own in every guise it has been protrayed. ST:Voyager was no different.
Federation space always gets bigger and there is always more to explore. There was no suggestion that there was "nothing left to explore" on the borders of Fed space at
any time in any series. That was never, ever, the issue with Voyager. Indeed, as Federation Space grows, the borders of Federation Space get bigger, not smaller, by definition. There is, literally, always more to explore.
In this specific instance concerning the new show being discussed, there is no issue of any conflict with an ongoing telvision show. DS9 is not on the air and neither is Voyager. The Star Trek movie franchise appears to be befuddled with unexpected delays. They should be shooting (or hell, have FINISHED shooting a sequel already if they were serious about making money with the movie franchise. Clearly, they aren't THAT serious about it.
There is no impediment against starting up a new series which returns to the premise of TOS and TNG, and just advances the clock.
IF there is anything they may decide to back off on, it is the idea of a Galazy class ship with hundreds of families on board. That was present largely as an excuse to justify Wesley Crusher's presence on the ship in an attempt to reach the tween and teen demographics with TNG.
Probably we can do without that element being reintroduced to the Enterprise F. Any number of reasons can be offered as an excuse. Or not, as the case may be. I don't think anything turns on it. In the end, the ship is the bridge, some hallways shot form different angles, a few staterooms, engineering, a meeting room, sick bay, transporter room and the Captian'r eady room. Really, those are the sets no matter how "large" the pretend ship is. From the outside, it's just a model with some more windows. In the script, its just a reference to how many peopl are on board.
It's alll fluff and no crunch.
In terms of tech, TNG already established the problem of the destruction of space time that high speed Warp was doing to the Galaxy. If you need the reason for a speed limit -- the canon readily provides such a voluntary speed limie (except in overriding emergrncy circumstances) already.
Frankly, there is no prohibition on them just restarting TOS as the TV series, either. Saldana aside, there is no actor in the movie series who has branched out into other films with a huge splash.
They could do a new Trek series on the small screen and make a good buck at it if they wanted to. Chris Pine, Zach Quinto and even Karl Urban are affordable in those roles for TV, (even if Zoe Saldana probably isn't.)
Not doing so is a preference, it's not a RULE.