If we get to a single payer system, expect to see what exists in the U.K., two tiers. That is, private health care for those who can afford it, public health care for everyone else. We have a bit of this today, here and there, some of it hiding in plain sight.
We'll see even more care delivered by non-physicians. Nurse practitioners and physicians' assistants will handle many routine cases, leaving the more puzzling or life-threatening ones to the actual med school grads.
As a result of such triage, expect to see the number of misdiagnoses increase, and perhaps fatalities as well. There's no free lunch, however much you might wish it.