In answer to the OP, GP practices are commissioned to provide medical care within 'core hours'. There is also commissioning of some Extended Hours. They are not commissioned/paid to provide services outside of this time.
111, MIUs and A&E are commissioned to provide urgent and emergency care outside of core hours.
GPs are neither saints nor overpaid lazy devils. They do a job, sometimes rewarding, sometimes very hard and/or thankless, do it to the best of their abilities, then want to go home, recharge and see their families, just like any other person. They are reasonably well paid: if they are Partners, they may earn very well but take on the risks of the business, and if things go badly, may earn nothing at all. If they are non-Partners, pay is reasonable but not generous and is considerably less than it would be in a fully private healthcare system with market forces allowed to act (and less than professional peers within similar length or level of study). For the amount of training (which is lifelong) required, disruption to life of the training process and risk of litigation in the job, they do not earn well.
Funding for GP is miniscule compared to all other medical services even though GPs provide the vast majority of daily patient contacts. And it has stagnated, whilst people are older, sicker and have very much higher expectations of what medicine can and should achieve.
It is a wonder than any young doctor choosing a training pathway chooses GP.
I hope your child is better, OP. I would suggest that 111 is probably the appropriate service, but not wrong to use A&E.