My current layperson's understanding is:
1) Redshift-to-distance conversion is a calibration based on the distance ladder.
2) And the distance ladder itself seems to have come under some doubt recently due to the "cosmological crisis" [1].
Can someone here ELI5 how they can be absolutely confident of these distances?
[1] "JWST just made the Crisis in Cosmology WORSE" https://www.youtube.com/watch?v=hps-HfpL1vc
The way I read it is that they're absolutely confident of their fantastic young age, that is, their redshift.
As you point out, the age-to-distance conversion isn't quite settled, but that doesn't really matter in this context I think, as younger will still mean further away.
So these galaxies will still be the furthest away (for now) even if the distance ladder gets adjusted "inward", ie closer to us.
At these extreme distances, I'm pretty sure any motion due to the local (to the distant galaxy) gravitational field is negligible to the perceived recession due to the expansion of the universe[1].
In physics, it is common for the writing to say that some fact was "shown" or "proven". Maybe look at the "absolutely confident" as similar!
I'm a skeptical and often dissatisfied reader of the fundamentals in physics: E.g., as a ugrad, I tried to be a physics major but too frequently got totally torqued off at the sloppy math so majored only in math.
To try to make a long story and explanation short, it appears that a common approach is just to go ahead and say "shown" and synonyms and keep saying that until there is some serious contradiction, conflict, new news, etc.
Still, "shown", ..., "absolutely confident" are not easy to achieve: It appears that we have a right to assume that such work did build on the accepted physics, got the arithmetic right, got the math right or close enough, etc.
And, there are some assumptions rarely explained: E.g., for these observations claimed to be from 13.5 billion years ago, we have to assume, maybe, things like the spectral lines of hydrogen, deuterium, etc. have not changed, the clock ticks and meter sticks mean the same as on Earth now, etc. Heck, maybe 13.5 billion years ago there was no hydrogen yet and dark matter radiated what we now believe came from hydrogen??? Here I'm making just wild stabs: A real conflict in the accepted results of physics would be big news and is not easy to find -- I can't give real examples here or at all.
So, for now, and until there is some big reason otherwise, we just let physics say "shown", ..., "absolutely confident". These are not nearly the worst ways the English language is sometimes used!