You have my sympathy.
My daughter had a bad experience on her first test.
I don’t think her instructor realised she was going to get a particular examiner as when he walked by her instructor said ‘that one is a misery’
She did get him and the examiner turned in his seat and stared at her and kept twitching near the brake. She felt so uncomfortable and it seemed to start a bit of a chain reaction. She hit the kerb when parking, and the test was cut short due to a ‘major’ fault.
Her instructor had described her as one of the best he had ever taught. Whilst we wanted to point out how uncomfortable the examiner had made her to possibly help others, we decided to wait until after she had passed another test.
Of all the bad luck she got him again. He was like a different person, friendly, chatty and much more helpful. She passed.
Maybe he had had a bad day, maybe she was being a bit too confident for a learner on the first occasion, we’ll never know.
The additional stress caused by the backlog in tests makes it even more nerve wracking as so much rests on those 37 minutes.
In answer to your question about whether to complain, I would say no.
Even if it isn’t it sounds like ‘sour grapes’ it can’t change the outcome, it will cause you even more stress in the long run, and would make things awkward if you got the same examiner again.
We used ‘Driving Test Now’ app, enabled auto re booking, and my daughter was lucky enough to get a second test 5 weeks after the first.
Good luck.