The only exceptions have been dentists and dermatologists - where I'm largely paying out of pocket anyway.
I'd absolutely go get extensive blood work done on a 6 month cadence and get covered PT if my cost approached zero.
And if your docs don't approve it and you want it anyway, pay out of pocket! I don't see how this would be detrimental to a single-payer system.
You would be a statistical outlier. People don't go to the hospital for fun.
There are lonely people who call QVC just to talk to the salesperson and end up buying whatever they are selling at that time. Those people also end up in the hospital and may realize it's cheaper if their copay is tiny or non-existent.
That said, I don't think it's limited to statistical outliers. The average, middle-class, insured family[1] is already likely to go to the emergency room even when they don't need to, causing long waits and drastically raising costs. Single-payer _may_ help with this (assuming there is much better communication about when to go, where to go, better scheduling tools, nurses available on the phone to mitigate emergency room visits, etc), but it's not a given.
[1] http://www.slate.com/articles/health_and_science/medical_exa...
Their system is completely overloaded due to their health care incentives.
I.e. What CA Single payer is proposing.