I work for Ambulance service and have seen this situation played out many times.
Very few paramedics have minor illness training, even fewer have a qualification that would give them credibility in diagnosing it. I'm 7 years in and only just done a minor illness course! Hence the need to speak to a doctor. Dealing with minor illness properly is actually far more complex then dealing with something like chest pain/heart attack - in terms of diagnosing and providing the correct treatment.
In a lot of instances paramedics would not be able to leave scene until spoken with a doctor. This is a medicolegal thing - if I see your child or elderly relative for example and say "well doesn't look that unwell, maybe you're right just needs some antibiotics, speak to a doctor when you can" , and leave - then the callback doesn't happen or takes too long, or something is missed and that patient dies x days or weeks later that's going to be on my head and also negative for the service.
Obviously this ties up crews for hours and reduces capacity to respond. However not having resources to respond to people is technically fine legally. Complaints come in all the time for "no one responded to my partner having a heart attack for 18hours then he died in surgery" - no one is even in a little bit of trouble for that as the delay was due to demand and nobody to send. Its just normal now. Last I heard 500 people a week were dieing in the UK due to delays in response and for treatment. This is why we are striking to get services changed and better supported so it stops overwhelming our ability to do provide any kind of proper emergency service.
We've got more ambulances and staff than at any point in history, but staff that become well qualified and experienced able to manage details effectively in a time efficient manner, having the knowledge to be able to confidently with standing tell the patient/family they can wait and try again with the gp or just don't actually need any treatment it's something that will get better itself - they get poached by private/other services who pay much better!
Whilst the rest of the staff are all being tied up dealing with minor illness, waiting to handover people who could have made own way, waiting for gp callbacks. I worked a night-shift last night on front line Ambulance responding to 999 calls. The only medication I used was patients own paracetamol they had in house but hadn't taken, not one person required any support to walk about the house, could have easily driven themselves to hospital or taken the bus actually if they actually felt they needed to go. But were calling 999 effictively because they wanted a checkup and it's more convienent to just call 999 knowing we eventually come and it avoids the difficulty of actually trying to see their gp. But we are not GPs, we don't not have anywhere close to their level of knowledge and experience in dealing with a minor illness presentation and differentiating it from something major, could very easily miss a diagnosis of cancer that would have been obvious to a GP. So 111 or similar keep sending us to check it out, but we are checking out things that we actually have no training in at all!