Why do Indian bodyshops not flood the UK, Germany, France, the
What is so unique in this combination of the US job market and India
that makes this phenomenon persist?
You touch on a worsening problem that apparently very few people have noticed, and which merits scrutiny. The least harmful effect intermediaries --from India and elsewhere-- cause is the false and misleading impression that jobs (as reflected in labor statistics) are being "created". Indian intermediaries abound in Canada as well, and they are not unheard of in Europe. But whether or not these agencies originate from India is irrelevant to the actual waste of time (and energy) their fake "screening processes" cause to candidates.
The problem is not just the mere flooding of so-called "consulting" agencies, but also --and more important-- that they usually post job advertisements that don't exist. That is why these agencies hardly ever provide any specifics about the alleged project or job position for which they approach the candidate.
There are tons of quite competent German and French speakers in Russia
and Eastern Europe, but they do not make unsolicited calls to job
seekers in Germany and France, to tell them about "an urgent position
with their client."
Non-Indian intermediaries and fake ads about "an urgent position" are rampant also in Europe. For instance, it is common for "positions" allegedly located in continental Europe to be advertised and screened by agencies located in the UK. In and of itself, that combination is not an unequivocal indicative of being a fake ad. However, the idea that employers in continental Europe would skip the multitude of nearby intermediaries and approach an unknown "recruiter" from far away is bizarre.
My experience with intermediaries from Europe is that they often try to gather information from the candidate as to where else he has recently applied, what those other projects/positions consist of, and so forth. That is the intermediaries' attempt to figure out what customer(s) to approach and offer staffing services.
Edited to add ...
One possible reason behind the combination you mention (US job market and Indian agencies) might be an intent to abuse the U.S. visa & immigration system. For years I have been hearing about Indian recruiters (for instance, Infosys) being investigated/charged for visa fraud.
To sponsor a foreigner for a work visa and/or an employment-based [U.S.] green card, the sponsoring agency/employer needs to prove to the Department of Labor that there are no qualified candidates in the U.S. market to fill a position.
By posting advertisements and/or conducting unsolicited interviews such as the one you found on Quora, the Indian agency might be trying to fabricate evidence that its efforts to find qualified, [U.S.] domestic candidates were fruitless. But this is just my conjecture as I revisit your inquiry. Only an investigator or an honest insider could speak with certainty about this issue.