But it's simply not true. my mom broke her back last year and before we realized that her back was broken, we saw an urgent care doctor, same day, within 30mins (I don't recall the exact timescale now, but it was pretty much instant). Who promptly gave my mom an Rx and told us to go to the ER. Personally, urgent care appointments have always been available within 2hrs, and even stuff like an xray (usually in a centralized office, so some travel required) is possible same day. This isn't special treatment.
If someone is waiting long for care, it isn't a problem with the system - and the alternative you speak of isn't going to solve the "I'm not a medical doctor" problem either, which is the main objection to the long wait-times. Minor hypochondriac-ness notwithstanding - nothing will be able to solve that completely - money is the back-pressure mechanism to avoid waste of limited resources. Whether anybody likes it or not, doctors/xray-machines/etc are not infinite (for now heh).
For-profit health-care insurance companies should burn in hell, though.