I would agree that if a site is science-heavy and requires a more nuanced scientific understanding, a scientist-practitioner model would be preferred and could be stated as such (is that an option, though?), and it also doesn't actually exclude an entire type of degree across the board (there are a few PsyD programs with this model, and both clinical and counseling PhD programs have scientist-practitioner training). In my opinion, this seems more reasonable than excluding across the board by type of degree.
EDIT: I think it goes back to what
@neurocog was saying about case-by-case basis. Some of us would argue that it's okay to state a clear requirement of X degree, and some of us would suggest that it seems like this is a sweeping generalization.
I would also agree that I'd rather folks weren't wasting money on apps that will get thrown out, but I wonder about how the underlying beliefs about the training were formed for some of these sites. I think it's worth asking the question if these particular sites operate via bias based on hearsay or they had a reasonable sample size of folks from X type of program that didn't work out well. There's no way for us to know, unfortunately, which is part of the problem.
There's also a lot of within-program heterogeneity as you mention, which complicates the issue.