You're right, that was a basic question and she should have answered it. It's fine to explore the reasons behind your asking it... but only as long as she also gives you an answer about how much experience she has. That's literally a textbook example of a question that even a "blank slate" therapist is supposed to provide an answer for, because you have the right as a client to know about her professional qualifications.

My therapist is a psychoanalyst, and is fairly neutral on the whole, but he never would have hesitated to answer a question like that--he's been very straightforward about his past professional experience, education, and training. I wouldn't even count those subjects as "self-disclosure." Her refusal seems obviously about her being insecure about her inexperience, not her theoretical orientation as a therapist.