The biggest issue is the lack of time/money for early treatment of issues.
Lots of things yet worse if untreated or make recovery more difficult - knees and hips especially. If someone can barely move for a while they get muscle wastage which makes recovery more difficult.
Also shoulder/back stuff. Quick referral to a physio, couple of treatments and back to normal vs 4 weeks wait for Dr, 4 more weeks for physio, sore shoulder pretty well locked up, needs more physio and possibly injections and therefore costs more to treat before you consider time off work and mental health implications of ongoing pain/inactivity.
Fortunately for minor physical stuff my family can afford to self refer to private physio for anything muscolo-skeletal.
Outside major cities there isn't an equivalent option for medical issues - no private GPs! If we did have access to one we would have paid to assess 4yo who isn't gaining weight (very active and getting taller so not urgent but 0th Centile for BMI so shouldn't totally ignore) and would probably would have paid to get 1yo seen for our own reassurance re constant colds, occasional wheezing etc.
The other hole is urgent but non-emergency stuff - fractures/bad sprains, cuts that might need probing/stitching or stuff that could wait a few hours but not until Monday morning from Friday night.
Fix the big holes in provision and access to primary care, fund mental health and elderly care properly and the A&E problem would almost certainly go away!