The question of recession velocity vs distance for distant galaxies that the question is posing is not very different than the continuing uncertainties in measuring the current Hubble constant, $H_0$. The former (velocity vs distance) is problematic because the distances measurements to far off galaxies depend on the distance ladder, i.e., determining cosmological distances accurately, whereas the second depends on distance measurements as well, but also on the cosmological model as to how the acceleration/deceleration changed over cosmological times, and the dependence of that on the relative mass and energy densities - i.e. they have some model dependence.
What is fairly clear though is that the dark energy started having enough of a noticeable effect about 5 or 6 billion years ago, based on distance measurements using the distance ladder (cepheid so to supernova and so on). So it is known, from measurements which are semi model-independent (somewhat because little in astrophysics and cosmology can be, you still depend on 'known physics' to get the distance ladder, and sometimes on simplifying assumptions), that the recession velocities are indeed time, or z dependent, and it is modeled by how the scale factor grows when the universe was radiation dominated, matter dominated, or more recently becoming dark energy dominated.
So the answer comes back to how non-model dependent can one measure cosmological distances. The fact is that there are still significant uncertianTies in the measurements of the Hubble constant using the astronomical ladder distances, and using distances estimated by the CMB-related/estimated acoustic density perturbations, and the time delay measurements of gravitationally focused multiple paths in CMB observations. The first measures about 72 Kms/sec/Mpsec, while the latter is around 67 Kms/sec/Mpsec.
Two interesting articles are on the Hubble constant measurement, the first more historical, the second more on the current Hubble constant uncertianTies, where both elucidate how measuring cosmological distances are at the heart of the issue, and the model dependencies. I've seen papers on the issues/uncertianTies of the distance and velocity measurements for different cosmological epochs, and don't remember them to be extremely more accurate, nor to be totally model independent. I don't see that, and neither do cosmologists/astrophysicists, as a big weakness, but more as the normal interplay of theory and measurement in physics. Still, while it is true that we still don't a totally accurate and certain picture of the expansion, any more accuracy in the measurements will refine the models, or find some new physics.
The two references are:
-A quick reference to the standard cosmology equations showing the model dependencies, at https://ned.ipac.caltech.edu/level5/Peacock/Peacock3_2.html
-A reference to the distance ladder, but not much on the most recent attempts to go further, in https://en.m.wikipedia.org/wiki/Cosmic_distance_ladder