While it is a problem, it's not a HUGE problem. Nobody enjoys waiting in the ER, so they only tend to go if they really feel they have to. I'll admit, I've gone many times to non emergency stuff, but often it's the only way to see a doctor on the same day. Some things may not be life threatening urgent but still urgent enough that you can't wait 3+ weeks for an appointment with my family doctor. They treat people by urgency priority and not first come first serve, as well, so people with real emergencies still get in faster.
Personally I think the best way to deal with that is if hospitals had walk in clinics with stable hours that you could go to for non emergencies. Even if it was a short time, like say, 5pm to 10pm. As long as it's consistent. There are clinics around town, but they have all sorts of weird hours, and sometimes they arn't even open, for just no reason at all.
Our system is not perfect by any means and does have it's issues, but it beats having to take a second mortgage just to get a basic thing done, or having to pay what is practically a mortgage payment per month on medical insurance.
One thing I really do think that should be fixed is the service should not be based on where you live. We all pay taxes. Everytime I hear about a service that is offered, it's only available in the big cities. For example, you can actually call a number and a doctor will come to your house! But of course it's only a few select cities in the GTA. We all pay taxes, if they're going to make a service like that it should be available everywhere.