Some opinions.
1) Federal labor laws prohibit using unpaid interns for a job that a paid person usually does. Because one has to prove that the unpaid person is doing the work of a paid person, the issue of being licensed comes into play. Medicine has solved this by granting licenses before residency.
2) CMS is part of the Department of Health and Human Services. CMS was created in the 1960s. Like a lot of bills, the law creating CMS had a bunch of other things. One of these things was the creation of a fund to pay for medical residencies salaries (i.e., Direct Graduate Medical Education funds). That Graduate Medical Education fund is administered by CMS. Because CMS is paying a flat fee for resident's salaries, they created rules to prevent double dipping (i.e.,, paying for resident's salaries, and then also being billed for each thins done by a trainee).
3) It gets a bit more complicated. CMS has both medicare and medicaid. Every year, CMS is given an annual budget. Then they have to divide up how that pool of money is distributed. Medicare is administered at the federal level (i.e., same rules in each state). Medicaid is federally funded, but administered by each state (e.g., CMS gives Delaware $500MM to run their medicaid program for a year, Delaware's medicaid office makes up their own rules). So why medicare might not pay for stuff billed by trainees, some states' medicaid programs might allow training institution to bill for trainees work.
4) When psychology is added, it gets even more complicated. Psychology doesn't get any of that sweet sweet CMS Direct Graduate Medical Education funding. Instead, the Department of Health and Human Services gives us some funding under a different branch (i.e., Health Resources and Service Administration). This pays a lot less than CMS.
5) Ready for it to get more complicated? By default, ALL providers in the USA are CMS providers. You literally have to write CMS a letter saying "I do not want to be a CMS provider". So we are bound by CMS rules.
6) Now we add in private insurance companies. So long as there is a licensed professional that is supervising, they can allow trainees to bill or not.
7) The VA gets their own funding for trainees, and they have their own laws. I don't know how that works.
8) Putting it all together: Psychologists are usually governed by CMS rules, but we don't get CMS money for training. Usually we cannot bill for trainees services. Unless they are in a state where medicaid says they can. Or if they are in a state where the state law supervisors can bill for trainee services, and the insurance contracts say they can do that. But usually that isn't the case. In order for a trainee to fix this problem, they would have to prove the supervisor is billing for their services, and that the trainee has replaced a paid person.