IMO, the problem really stems in the types of ways interviewers go about trying to assess skill sets. Rather than assess for the very specific skills that will be needed in order to do the job being applied for, it becomes some random take-home exercise they likely copied from somewhere. I take issue with the assumption that devs are against any kind of skill assessment. They are just against skill assessments that do not reflect the skills they will need to be successful in the job at hand.

In my experience interviewing, a properly designed technical interview exercise can fairly accurately peg an interviewee's skill levels in about an hour, as well as their ability to communicate with the team (taking into account nerves will likely reduce their performance in an interview setting). I think having a take-home exercise could be an option that could be offered, particularly if someone is struggling in an in-person(or video) interview session, or even something that is flexible ahead of time to allow the interviewee to put their best foot forward by choosing the evaluation that best suits their strengths.

I do agree with your point that interviewers need to have an ability to spot "good interviewees" - some people are just really good at interviewing. I don't think that automatically means take-home assignment though, I think having gradually more specific questions to suss skill level out (which you mention, i.e. "better questions") is really important.