Interesting.
I've always been in psychodynamic based therapy, as it works well for me.
I did however spend 6 months in counselling with a private CBT based psychologist, who was however open minded and pretty eclectic. In fact she advised referral to a Kleinian analyst, in the first session with her. I declined, because of my lack of knowledge at the time. But, looking back, I see how wise she was.
I'm lucky in that I have financial support for private therapy.
People that rely on NHS referrals have less freedom of choice.