Pay more for a visit to the doctor?

This issue is under discussion - or should I say there are howls of rage?
Downunder has a Medicare scheme. A levy is imposed on all tax payers to fund a national health scheme. As a result some people get "bulk-billed" for a visit to their GP - i.e. they pay nothing - and others have to pay. Those who pay are of course those who earn an income over a certain amount. There has never been a requirement for doctors to "bulk-bill" anyone and some don't. 
Bulk-billing however has led to two things. The first is that some patients have taken advantage of or even abused the system and made many more trips to the doctor than are necessary.  The second is that some doctors have abused the system by seeing many patients - perhaps one every ten minutes.
I suspect that the doctors who do this are very much in the minority but it may be a different story with the patients. I know people, particularly elderly and lonely people, who see a trip to the doctor for some minor ailment as a social outing. I know other people who will go to the doctor seeking antibiotics for a cold, because they have stubbed a toe wandering around bare-foot or because they have been bitten by a mosquito. (Yes, really.)
Then there is the "sick-note" so you can "take a sickie" (sometimes a day off to recover from a hangover) and "doctor-shopping" to find the doctors more likely to grant you these things and the antibiotics/attention you crave.
And now there are suggestions that, should the system be changed, the emergency departments of hospitals will be overloaded with non-urgent emergencies.
So, what is it with us and doctors? The Senior Cat needs to see the doctor once a month - because the government will only allow a certain prescription to be filled for a month. Someone, somewhere - when the drug was new - decided that the patient should be monitored once a month. Years later that has not changed although both doctor and patient know it is not necessary.
I try not to bother my GP. She knows that I will only see her if it is absolutely necessary. I have had one lot of antibiotics in the past thirty or more years - and yes, I was ill enough to need them. I have been fortunate.
But I wonder about other people. Are we getting sicker or turning into a nation of hypochondriacs? Are there a lot of lonely people out there? Do we need a different sort of medical system - one which would allow people to turn up to a "minor ailment" clinic where someone could take their temperature and blood pressure and listen to their chest and then decide whether to send them on to the doctor?
I would not like to be a GP - or indeed any sort of doctor. If I need to pay more to visit my GP I will - but I will only go and see her if I need to.