So, what exactly made private insurance appealing to doctors when it first came around? You could argue that it gets more people in the door and increases consumer base, but I thought that was the problem?

I think the idea is that in a completely free market, to the degree that a private insurance plan gets its pool of lives "in the door more frequently (increasing demand)", there would be a proportional increase in the premiums paid by the individuals in the pool. Of course, in a free market, individuals could choose a plan that sets limits on their demand e.g. plans that only cover unforeseen catastrophic events. This, by the way, is what insurance is supposed to do, you aren't supposed to "insure" against the "risk" of your yearly routine doctor checkup.

So anyway, with private insurance, the premiums paid -- while spread across a pool -- should reflect the amount of healthcare demanded. This is extremely important regarding economic efficiency. Note that "high costs" aren't necessarily "good" or "bad". To see this, think about what would happen and why it would be a bad idea for the government to mandate the price of all laptops to be $10 (or $15,000).

What matters is efficiency, and with private insurance there is efficiency because, like I said above, the amount demanded (healthcare) is voluntarily given up (higher premiums). This is one problem with single payer. It screws up efficiency. Single payer in healthcare is exactly the same as the government coming in and mandating all laptops be sold for $10 or similarly that they be sold for $15,000. This is central planning and central planning can never, even in theory (look up the Economic Calculation Problem), get the price "right" to achieve economic efficiency.

Also, there is tens of thousands of pages of regulations, hospital zoning laws, FDA restrictions, practitioner licensing, I could go on. Removing all those idiotic regulations would go a long way to reducing costs as they artificially restrict supply. Then to appease our uneconomic emotional citizens, universal healthcare (not single payer) in an otherwise completely freemarket with only private insurance is de facto just wealth redistribution to those who can't afford private insurance. This might cause some very minor distortions to healthcare market/costs but probably not much. Of course, there would be some less minor distortions elsewhere from the redistribution itself (e.g. deadweight loss from taxation), but not specific to healthcare.