It’s only a contradiction if you ignore the parts that you cut out.
You will never observe a star that has moved out of the observable universe.
Once again, the part that you cut out explains what you are missing.
A star that is a billion light years away is slowly accelerating, slowly increasing its redshift. A star that is 2 billion light years away is accelerating away faster, its redshift is increasing faster. A star that is 4 billion light years away is accelerating away even faster than that, and its redshift is increasing even faster. The further it is, the faster it is accelerating away from you.
As it approaches the horizon, the acceleration approaches infinity, as does the redshift. Once it crosses the horizon, it is just gone.
You keep seeming to want to think of this as something accelerating away from you, approaching the speed of light. In that case, what you say holds true. If a rocket with a magic engine maintains a 1 g acceleration, then after about a year, you will see it as the clocks on it slow down more and more, the light from it becomes more and more redshifted, and it asymptotically become more redshifted and slow. But here’s the key to that, if you measure its acceleration away from you, you will see that it is not maintaining 9.8 m/s/s, but its acceleration itself is slowing as it approaches the speed of light. To someone on the craft, it feels as though they are still doing 1 g, but to the observer that they have left, they will be accelerating less and less as they get further from you and recede faster.
This is not the case with the cosmological horizon. Objects are accelerating towards it, and don’t slow down as they approach it. They keep speeding up, until they cross it, and they cross it because they are now superluminal compared to your frame of reference.
The asymptote here is time, not distance or velocity or acceleration. If an object is going to cross the horizon at t=1, then as t approaches 1, then that is when you see observe object get redshifted away. And if you mathematically observe it as it gets arbitrarily closer and closer to 1, you will never actually see it “disappear”, it will just become more and more redshifted.
But, just as in Zeno’s paradox, the answer here is that time itself does not slow down, time itself does not get arbitrarily closer and closer to 1, it actually goes to 1 and then keeps going. Once you have gone past 1, the object is no longer observable by anything that travels at C or slower. (It is observable by tachyons, the only problem there is that tachyons don’t exist.)
Put it this way, it’s not just moving away from you, you are moving away from it as well. If you are moving away from it at greater than the speed of light, then how is a photon or a gravitational wave going to catch up with you?
And as t approaches 1, λ approaches infinity. Once t has passed 1, there is no longer anything there. You keep forgetting that space itself is expanding here. This is not an object that is undergoing acceleration to take it away from you, it is an object that has more space between it and you than it can cross before more space is made. It can never catch up.
As an imperfect but useful analogy, you can use water. If you have a flow of water that is traveling faster than the speed of sound in water, then if you are upstream, disturbances and waves cannot propagate to you.
No, I didn’t, and you are correct that it is irrelevant, as in neither case does it matter. Once it crosses the cosmological event horizon, it is no longer observable by any form of propagation that travels at the speed of light or slower.
