I'm in my 30s and have met mannny therapists over my life; I know we're not in the same country so there's differences, but in my experience private therapists can often be more attentive and respectful, have more time to listen and focus on you as an individual, and be a lot less likely to push through a checklist or funnel you into categories of "fits this diagnosis exactly".
There's still an aspect of luck when you're meeting someone new, of course, but I've consistently found private therapists to be better in quite a few areas. I have an amazing therapist now who pays attention, so I don't have to say the right things or explain everything perfectly; if I'm abruptly speaking less he knows I'm distressed, if I'm looking away a lot he knows I'm more disassociated.
All this to say, you shouldn't have to be perfect at expressing yourself; if I was a therapist I'd expect that someone would have a harder time verbalizing when it comes to certain topics. What I really want for you is a therapist that focuses on you as an individual, gives you time to get to know them and feel comfortable, and supports you while letting you lead the conversation and bring up topics when you're ready.
Can you approach the private therapist appt with an open mind, and while not deciding 100% you have to tell her about the system beforehand (as I understand that's a huge deal), feel it out and see if she's someone who you could feel comfortable with?
Any therapist worth their salt will be understanding if you say there's some things that are hard for you to talk about. And will want to spend the time it takes for you to be comfortable sharing big things with them. If they don't, they're not worth their salt (hmm, is that a saying in the UK? I hope so, otherwise you're going to wonder why I'm talking about salt
![:)) :)) :))](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
)