Cool. So while most antibodies aren’t catalytic, some are and sometimes that activity may be useful.
Stepping back to the broader question, I see that there is actually a substantial literature about catalysis by small peptides, which occurs quite often. For example, one paper (https://pubs.acs.org/doi/10.1021/ja0108584) reported construction of a library of octapeptides in which six amino acid positions were randomly populated from 14 amino acids. Roughly one thousand of the resulting octapeptides were tested for the specific catalytic activity they were interested in, and they found multiple ones that worked.
So there does seem to be abundant evidence that catalytic activity is not particularly rare among peptides. Now, stepping back to the original question, is there any evidence at all that such activity actually is extremely rare? Axe’s study certainly doesn’t provide it, since he was only testing variants of a single catalyst, which provides no information on the probability that unrelated proteins and peptides would have a similar activity. Why are you so confident that it’s rare?