Here's the thing, their claims about racism in hiring can be true, and yet their experiments still not "show" that those claims are true. It is critical to keep these things separate. If the methodological problems are real, then by definition, the studies do not "show" what they claim, because that would require that there are not plausible (and in this case probable) alternative explanations. Do you think that some employers are less likely to give a call back to a non-native English speaking or foreign born applicant? Do you think that some employers would sometimes read an unfamiliar name rooted in a foreign language in infer (whether consciously or not) that the person might be non-native speaking or foreign born? If your answer to these questions is "yes" (and rationally it should be), then the logically neccessary conclusion is that the results observed in these studies would have emerged even in a world without anti-black discrimination. Data only "show" that a claim is true, if the data would only have plausibly emerged in a world where that claim is true. Thus, these data do not show the claim is true, despite the fact that the claim might be true anyway and that you might have other basis to believe it is true.
I think there is basis to think that there is racial discrimination in hiring, but I don't think there is a sound basis to believe that the prevalence of it has not declined at all in the last 30 years. In fact, there is indirect evidence that makes this claim implausible, such as evidence that racist beliefs have significantly declined during that period, that blacks have become more educated, with many more blacks holding positions of authority and respect, including on being elected and reelected president, which was unthinkable in the 1980's. Given that racist assumptions and feelings about blacks would be the mechanism behind hiring discrimination, this makes it implausible that hiring discrimination has not declined.
In contrast, what would be less likely to have declined over the years would be employer motives to not hire non-native English or foreign born applicants. Which makes this mechanism not only plausible, but a more plausible explanation for the lack of change in the size of the difference in callbacks observed in those studies over time.
You have other reasons, perhaps very good ones, to think that employers have a bias against black applicants. That doesn't mean that racial discrimination is the best explanation for these particular data, particularly the lack of decline in the observed result, which is the main reason you posted the meta-analysis to being with.
Other than things I have already offered, some more points in support of what I just said might be, for example, (a) that I read that the use of 'ethnic names' has supporting prior research to suggest that it is a useful indicator of the relevant attitudes,
That research merely shows that if you ask people to decide whether someone named "Lakisha" is more likely to be black or white, most people will say "black". But the baby registry data also show, that only 1 in a 1000 blacks are named Lakisha and only a small % named with other such race-signalling names. Which means the names signal being a special non-representative sub-set of that race. Plus, nothing in that prior research addresses that those names also signal other things only incidentally correlated with race, such as language skills, education quality, cultural background, etc.., and they are just much less familiar words and simple lack of familiarity produces subtle negative feeling like anxiety that could easily tip the scale in a some cases.
(b) that even if the 'ethnic names' resumes is not a perfect approach, it still has a lot going for it, not least the amount of control that can be exercised over all other aspects of the material used, (c) the 2 studies I have read most closely do seem to be pretty rigorous, such as in questioning their own limitations (although not the main one you have highlighted) and trying to correct for things like possible publication and write-up bias,
They are rather dismissive of the limitations they briefly mention, and then go right on making strong causal conclusions. And the limitations I'm bringing up would be rather obvious to any expert in the field, so their failure to deal seriously with them suggests a deliberate effort to avoid noting the more serious flaws with their methods.
(d) the results of 'in person' alternative type of studies, which of course may suffer from slightly different potential limitations, do seem to corroborate the 'blind resume' studies, (d) despite all that you say (and I have noted it) if we accept that 'there is racism' I still think it would be overall slightly odd if it was not going on in this sphere of human activity,
This relates directly to my first point above. You are using other information and data outside these studies to support the general conclusion of racism in hiring (which is fine and rational to do), but then using that to decide that these studies empirically show that this conclusion is true (which is not valid to do). What the studies empirically demonstrate and what you think is true due to other information are separate things. And note that even if you think racism would have to have some impact on hiring that doesn't support the conclusion of this meta-analysis that this impact has not changed in the last 30 years.
and finally, (e) even if the 'African-sounding' names do trigger a (possibly automatic or subconscious) response, and even if in some cases this prompts the reader to think 'possibly foreign', this may still at the very least indicate some sort of 'othering' bias
Sure, but it's a very different type of "othering", especially when it conjures the ideas that the person may have limited proficiency with the language essential for them to perform their job effectively.
and since the names are African-sounding and not for example Irish- or Russian-sounding, it may still be indicative of a racial bias in terms of a particular ethnicity.
Sure, this might be true and it might not be true. The studies do not show either way. They merely provide data consistent with that hypothesis but also consistent with several other hypotheses for which we also have other prior information to suggest are true. My ability to predict someone's gender from their name is consistent with me being psychic, but because it is also consistent with other highly plausible explanations, it does not show I am psychic or even qualify as evidence of it. And while you have more a priori reason to doubt that I am psychic that is a separate issue from when the data in question show.
Also, you just unwittingly hit upon a very simple method that would vastly improve these studies that none of these researchers has bothered to try all these decades: Simply make the "white" names foreign sounding names from predominantly white countries like Russia. What does it say about the competence of the researchers in this area that in 30 years, virtually none (and maybe truly none) have even bothered to mention this issue of the foreign/language implications of the "black" names, let alone bothered to improve their method by using "white" names with similar foreign/language implications?