I would agree with everything said and recommended above. In my experience, it's not unusual for internship sites to be stricter than many graduate programs when it comes to fidelity of test administration, and thereby requirements to test out for independent administration. Internship supervisors can also be a bit more risk-averse in this area than are some/many grad school supervisors. This can then feel like you're taking a step backward. The internship site has just likely seen variable proficiencies in terms of trainee's testing experience and abilities, even in those with hundreds of hours of testing, and may have been "burned" by this in the past. I certainly wouldn't take it personally, and I think the suggestion to ask for a definition of "success" and concrete anchors is a good one.
Personally, I don't think a trainee needs to have every script of every test memorized verbatim; I don't, and I wouldn't expect that of a trainee. But for those tests they don't have memorized, they do need to be able to use the manual/testing materials to administer the test fluidly and, afterward, to score it correctly.