This is actually the subject of an upcoming US Supreme Court case that will be heard this fall.
The State of Minnesota sued Mayo Clinic because Mayo was not witholding FICA taxes from their residents. Mayo's defense was that residents are students, not employees, so don't have to pay FICA tax. The 8th Circuit court found in favor of Mayo, so the State of Minnesota has appealed the decision to the US Supreme Court.
This case has major implications for residency protections and benefits. If residents are reclassified as students, instead of employees, we can lose a lot of protections. Our status as employees gives us protection from wrongful termination, ADA protections, health insurance/disability insurance, etc. Students have no such protections, and this would open us up to even more potential abuses from hospitals.
I agree that in theory it would be really nice to get back the additional $3k-$5k in FICA taxes each year. But in the long run, I think the legal protections and benefits that come with employee status are much much more valuable than having that amount of money in my pocket.
My understanding is that the AMA has actually filed an amicus brief with the Supreme Court, advocating for residents to continue to be classified as employees, for many of the reasons cited above.