> Won't a photon climbing out of a huge gravity well have a huge redshift, thus confounding estimates of distance from us and estimated age?
Of course it will. It will also have a redshift related to the expansion of the space it's been travelling through. Both of those corrections are absolutely part of the model. It's not nearly as simple as "Astronomers forgot about General Relativity!".
But I guess it's true (to be clear: I'm just an amateur in this field) that the ΛCDM model for cosmological evolution that we've all been looking at for the past decade or two isn't holding up well at all. It looked like it was pretty much there and just needed some fine tuning. Then we got a bunch of new data and everything's a mess.
That's kind of exciting all by itself, though it's also leading to a bunch of nattering from the existing iconoclasts (MOND nuts in particular) whose theories are also not working very well to explain JWST observations.
New insight needed, basically. We're all watching for updates.