That hasn't been the case with the tax-based healthcare we already have (Medicare), so I don't know why we should assume it would be that way if Medicare was expanded to cover everyone.  Medicare actually covers many innovative procedures BEFORE private insurance will cover them, and in many cases, AFTER private insurance stops covering them due to these expensive procedures putting a crimp in the greedy insurer's profits.  (Hip resurfacing is a true-life example of the former; gastric bypass a true-life example of the latter). 

And don't forget, there will ALWAYS be private insurance available for those who wish to have access to more choices than under Medicare or the like.    Of course, it's true that treatments/medications would need to be "evidence-based" under a governmental system.  Nothing unproven, experimental, etc.  But private insurance doesn't cover that, either.  Also--private insurance almost always has a lifetime maximum that it will pay out in benefits per subscriber, unlike Medicare.  I would choose Medicare over most private health insurances any day!