What is the real problem(s)?

The real problem is the triad I mentioned above. To be politically palatable, a conversion to single-payer has to:

1) Give nearly everyone access to above-average health care coverage (or else the people whose coverage gets worse will be pissed); and
2) Keep provider compensation more or less at current levels (or else the millions and millions of workers in the health care industry will be pissed); but
3) Not make people pay more for health care in taxes or premiums than they do already (or else they will be pissed that they're paying more).

That's the problem that has to be solved in order to make the switch. It's not a problem with the current system (of which there are many) or a problem with single-payer (of which there are still some). It's the barrier to making the transition. There actually isn't a way to deliver all three things. But because you have to break one of those three things, it's politically impossible to get it done.

It's more satisfying to blame some group of unpopular entities (insurance companies or 'big pharma' or whomever). But the reality is that if you want to reduce the amount we pay for health care, you have to reduce the amount we pay for health care - and we don't spend enough on "wasteful" things to do it just by cutting waste.