Thank you all for the replies! I am very grateful. I decided I could not work with someone like that. As many of you pointed out, it feels too rigid and life does happen. I think giving an advance notice for not being able to make an appointment is reasonable but not being able to do that is certainly not. I would not have even met with her, had I known about her policy. In a way, I feel she purposefully did not mention that until I made it to her office which is even more bothersome (at least now I know to explicitly inquire about this before I meet another T). She does come from psychoanalytic background and that was in part her justification for the policy. I have called a few Ts around here and it seems that the good ones are all booked weeks in advance and/or have waiting lists. So the fact the she did have a few available times (not a lot) despite her rigid policy, perhaps speaks for itself about her ability to retain clients and ultimately the quality of her work.

I asked the question here because I was uncertain whether this is a common practice as she implied it was. As far as I could tell that doesn't seem to be the case.