[codethief]

> Won't a photon climbing out of a huge gravity well have a huge redshift

Yes. Though see the sibling comments on why the (remaining) gravity well the emitted photon has to climb out of is not particularly huge here.

> thus confounding estimates of distance from us and estimated age?

Generally speaking: No, because people are not stupid. :) See the integrated Sachs-Wolfe effect for a very similar situation (CMB photons traveling through gravitational wells) -> https://en.m.wikipedia.org/wiki/Sachs%E2%80%93Wolfe_effect