America fares badly in a comparison of health measures in rich countries

If you are arguing that geographical distance leads to inefficient supply lines, this seems dubious. Australia, third from the top, has the lowest population density of most countries listed, having a land area almost as great as the USA, but with one fifteenth of the population, and with by no means the same level of transport infrastructure.

Given that the US is aiming to inoculate near 90% of the population for the current influenza problem, your deductions about distribution may need to be examined more closely.

Italy + France + Spain + Germany + UK + The Netherlands + Austria + Sweden + Portugal + Norway ==> 360 million people, way more than the USA.

Maybe the USA are just currently not very good at it.

But it's alright to be subpar, given the amount of money sunk into health care in the States they can get to the top given the proper reforms and some time.