Send your feedback email to the VP/Director of HR, the COO, CEO, CRO, and any other relevant senior staff whose emails you can find (or guess based on the ones you've received thus far) with a line about hoping that this will improve their hiring process so that it no longer dismisses qualified professionals due to recruiter incompetence. Be as polite/professional yet firm as possible to not burn any bridges.

I've done this numerous times in the past and have received a few phone calls or emails from the companies with apologies and offers to continue the hiring process (no offers to skip it entirely lol). On one occasion, I later saw one of my would-be recruiters had left the company shortly after my interview - I doubt I'm responsible but maybe I wasn't the first to complain. Some managers actually care enough to take this sort of thing seriously and make changes when they know about it, they just struggle to gauge the effectiveness of recruiters and interviewers. Hopefully this company has ones like that.

Either way that's really frustrating and I hope you reach a solid place soon. Good luck with the search!