Unfortunately, at least in some regions, there is no other place to receive urgent (but non-emergency) care. Waiting a week or more for a doctor's appointment isn't going to cut it if you've broken your arm or if you've been bitten by a possibly rabid animal, and so you're stuck waiting all day or all night in the ER.
It was a revelation to me when I moved from the US to Australia. Not only do I not have to pay for insurance, but it's so much easier to receive non-emergency urgent care as well. I can typically book a doctor's appointment for the next day (there are no insurance networks to worry about, so I have my pick of doctors), or if I need care sooner than that, my local ER usually has a wait time of under 30 minutes even for super minor treatment, like removing a sliver of glass from my foot (the high availability of doctors means the local is pretty quiet).
You're seeing public GPs or what kind of doctor? In Melbourne, public bulk billed GPs are hard to book for the same day, the best ones are regularly 1 week away. Private specialties (Endocrinologist, Psychiatrist) are months away. What you're doing that I'm doing wrong?
