I run a GP surgery.
There are definitley surgeries where patient feedback is not listened to.
We don't have set days or clinics as we understand that people cannot always fit with us and should not have to take annual leave to come in.
We have appointments put aside for the same day, the day after and 2 days later - urgents are seen the same day and routing in 3-5 days. It can be done. You GP practice is contracted to deal with you from 8am-6.30pm - just don't call at 6.15 and expect to be seen at 6.35 - you will be referred to out of hours our the GP would never leave the building!
We do have patients who do not want to speak to our reception team. Our receptionists are highly trained in GDPR and confidentiality - they do not diagnose conditions. However, they do sit there with a list of over 30 minor conditions the GP has told them can be seen by a pharmacist. The receptionist is trying to ensure the GP only sees patients that require a GP like cancer reviews, anxiety/depression, complications of chronic diseases - rather than ear infections, warts, shoulder aches that a pharmacist can see.
99% of our callers are lovely - but 1% are abusive, threatening, inimidating and condescending when trying to be helped.
However, to add some balance, I know of another practice where the receptionists are made up of 99% unhelpful/abrupt and 1% wonderful - Oh, how I would love to manage/sort that team!!!
OP, I hope you get the appointment you need very soon.