I would be really bothered, too. Not so much about not having a set time (I am a student with a really flexible schedule), but that the therapist did not hold to seeing me on a regular schedule in terms of sessions per week. Before COVID, I did not have a set time. However, my therapist never forgot me and that I would need a couple slots. Sometimes, she would say her schedule was getting full and she had saved a couple times for me that she hoped would work - and they were always times that my appointments were often scheduled. I could count on her to not let me slip through the cracks, so it didn't bother me that I didn't have a set time. However, in your situation, the lack of a reserved time would really stress me out since it doesn't sound like yours keeps clients that will need appointments on her radar.

Personally, I have a low opinion of therapists who keep more clients on their caseload than they can manage to see at the agreed upon frequency. I just don't think it's acceptable to be overbooked and inaccessible to clients that one has agreed to provide therapy to.

Sorry you're in this situation. Did she actually say she is unwilling to provide a set slot, or are you just assuming based on how she's operated so far? I didn't think mine did set slots, but something happened (I forget what it was), and she asked me if I wanted to just see her at x and x day/time every week.