So there’s a lot to unpack here, and I don’t think this will cover everything but here goes with a few things:
So, to start with, from your own point of view, you don’t see time your own time slowing down or speeding up. You experience one second passing as one second. That never changes.
Another is you don’t see time slowing down or speeding up for other people. If you sent a twin out to near the speed of light and the twin had a clock you could see, you would see the twin aging in sync with the clock, i.e. a year on the clock and the twin looks aged one year. What you may or may not see is the clock in sync with your own clock. Acceleration isn’t even relevant to that.
So, let’s set up a little thought experiment. You have two observers A and B in space that somehow start out traveling apart from each other at some constant velocity (less than c of course). No accelerations to get this set up. Each has a clock that can be seen from everywhere else.
First, which one is at rest and which one is moving? The answer: both and neither. There is no preferred frame of reference. You can say that A is at rest and B moving away. Or that B is at rest and A is moving away. Or both are moving away from some common point C. Any or all of those are equally valid ways of seeing things.
With that said, what do the clocks say?
From A’s point of view, their own clock is ticking away at 1 second per second. But they see B’s clock ticking ever so slightly slower.
From B’s point of view, their own clock is ticking away at 1 second per second. But they see A’s clock ticking ever so slightly slower.
From C (that point that sees them both moving away), they see both A and B’s clocks ticking ever so slightly slower than 1 second per second.
And these are all equally valid. And there’s no ‘correct’ answer as to which clock is ‘actually’ ticking slower. You could bring the clocks into the same place but that requires an acceleration. And even there, A can say that B accelerated, while B can say that A accelerated but each would claim a different acceleration vector, which would make all the observations consistent.
So, it is very important in these sorts of problems to specify where you are measuring from, i.e. the frame of reference.
The Hubble Horizon defines the boundary between objects traveling faster or slower than c from some given frame of reference. As above, the frame of reference is rather important. Objects near it can be at any sub-c speed or acceleration, relative to the frame of reference.
So, if we are defining that frame of reference as ‘Earth’ and then posit an object already traveling near c, we would see a clock on that object to be ticking slower than ours while we see our clocks ticking away at 1 second per second. But likewise, from the point of view of that object, it would see clocks on Earth as ticking slower while its own was ticking away at 1 second per second. There’s no ‘correct’ clock in this situation unless you define ‘correct’ to be ‘from my choice of frame of reference’.
And that’s how things work in the real world. We’re on the surface of a spinning planet, in orbit around a star, which orbits around the center of a galaxy, which is itself part of galactic clusters in relative motion from other intergalactic structures. Relative to something out there, we’re traveling at a terrific speed, yet we see our clocks ticking away at 1 second per second. We could estimate how fast/slow our clocks appear to be at that other frame of reference, but that wouldn’t really affect anything in our own.
Ages differently, anyway.
There’s no ‘true’ measurement of time. You can equally say that the solution to the paradox is that the twin that was shot off into space aged more slowly as say that the solution is the twin that stayed on earth aged more rapidly.
Both are equally valid, depending on your frame of reference.
But seeing as most of us are on Earth, it is natural (for humans anyway) to prefer to use Earth as a reference.