The thing that would most annoy me about that is that it would presumably be okay for the therapist to cancel because they are sick, away, had a family emergency etc.

I think I've only once cancelled at the last minute. My T has a 24 hour policy as far as I can remember. Once she was expecting me and I didn't turn up because she'd told me a different day than what she had written down. However she has on occassions double booked accidentally, or booked me in for an incorrect time and by the time she realised had no spaces left when I could see her. So honestly, if I'd messed up our times more than she had, no problem paying for the cancellation fee. I haven't had to, and as I've often accommodated last minute changes for her, it would bother me, if it was for a genuine reason that I had to cancel. So I definitely wouldn't consider seeing someone who would not let me cancel for any reason, but surely would have to cancel at times themselves. Even if they didn't cancel, I wouldn't particularly want them to see me when they were really sick, or in the middle of family crisis.