It's a review paper, and (mostly) about the different techniques used to untangle 'local' effects from universal expansion, in the analyses of large redshift (survey) datasets.

The author comments on the techniques used in several such analyses, in section 8 of the paper, especially comparing and contrasting the methods used.

I'd say, based on just a quick skim, that it doesn't directly discredit redshift quantisations, but that unless anyone who claims to have evidence of such redshift quantisations, in large datasets, has done their analyses with at least the same degree of care as outlined in the Hamilton paper, then any results claiming the existence of redshift quantisation would surely have a very shaky foundation.

Or, turning this on its head, as none of the ~20 separate analyses discussed in section 8 seem to have turned up any evidence of redshift quantisation, then it must either very subtle indeed, or occur at a level below the threshold of these analyses.

In any case, now that we have both 2dF and SDSS datasets (and many analyses), both of which came out after this Hamilton paper was written, we have even more compelling evidence for the non-existence of redshift quantisation.