If you can't see GP look out for a walk in centre or 111 - they often have GP hubs you can be booked into.
Yep it can definitely be really tricky to get an appointment alongside school runs etc. It's such an impossible situation to solve (other than masdsively increasing the number of Doctors/ACPs, or somehow reducing the number of people trying to get an appointment).
I work at a surgery for 50% of my role, there have been lots of trials. Ultimately you've got routine appointments that people will always attend/have to occur for routine monitoring and prescriptions to be issued, so these can be booked in advance. But your urgent/acute problems need to be same day - because if on Monday you open up the slots for the whole week bookable by phone on website, the whole weeks appointments will be booked full on the Monday. The rate of DNAs/missed appointments now massively increases since appointments booked on monday for friday are now forgotten about, no longer convenient time, or the problem they were calling about is already getting better - so loads of wasted slots Same day only reduces this from happening.
The majority of people I see coming in and have booked appointments are 'self care' issues which will get better without any treatment - I have 16 slots in the surgery + a few home visits, there's normally 4-5 cough for 3 days feels needs antibiotics (very unlikely), sore throat wondering if needs antibiotics (very rarely), ?thrush - advise to speak with pharmacist as otc treatment available, 2 or 3 bouncy happy children had a fever the night before can we check them out....
These are problems which really shouldn't be coming in taking away slots from people who do have more concerning symptoms which actually need a medical assessment/aren't obviously self care - unusual headaches, recurrent/persistent abdominal pains, weight loss/tiredness, lumps and bumps,....
So people always argue about they've paid their taxes, they should be able to have a doctor check out the child who's only symptom is a slight fever for a few hours. But with finite appointments if this person calls first it can often take away the appointment from the parent calling for their child who's had a persisting fever over a week not really eating or drinking looks unwell, who might actually need to be treated/further investigations.