A, an insurance company, is the one fighting to get the cost down from B, the healthcare provider.
> And if there were more smaller insurance companies competing with A, wouldn't it make sense for A to just eat them up to eliminate competition
Monopolization is more dangerous from the healthcare provider side, though it happens both ways. Basically, insurances compete against each other at state levels, while hospitals only fight the nearest hospital, maybe.
> And if there were several equally large insurance companies aside A (say, B, C, D), wouldn't it make sense for them to collude and keep prices and profits high?
You can see that health insurnace companies are not really profitable. They have lower margins than auto-insurance.
> And since A (a single large) or A+B+C+D (several large) companies surely have considerable clout (money gives you that) isn't it their natural interest to spend on buying politicians, media, etc, to make sure things get even more in their favor?
Noone has more interest than to implement national health insurnace than politicans. It's the biggest political boon plus almost unlimited spending they could get. If something like M4A passed, the state would grow in the order of 10% of GDP.
> ow would insurance being a private market make it "make it more efficient for the sake of maximizing profit"?
By keeping at bay provision costs vs quality. In Healthcare there is something called the Iron Triangle: Access vs Quality vs Cost. If you unleash access, you need to lower quality or increase cost.