Closer to $5k.

US in-patient costs are about double that of the European average.

An ICU bed that is $1,000 per day in Europe, is around $2,000 in the US.

But the hospital will have a charge master price of $10,000 per day.

Insurance company negotiate agreed charges that are often 80-90% lower.

That’s the number that contributes to GDP.

Yes that's the average spent per citizen which doesn't say that the brain surgery will cost just the double of EU. Many Americans apparently don't seek medical help if they feel like they can wait it out, hesitate when call an ambulance etc. Which is not a thing in Europe, its not something you give it a consideration from financial standpoint. If you are poor and you twisted your ankle, you call ambulance. If you have a rash that bothers you and you are poor and unemployed you still go to the dermatologist. If you are rich, you can seek private care(yes, many countries do have private hospitals where you can get an expensive care if you like).

Anyway, that's another topic and you are right that for the GDP purposes the spent per capita is a good metric. Which is still on the point, the same service costs at least twice as much resulting in useless numbers if you are trying to use GDP as a productivity metric(or anything humane, in fact).

We see a similar thing in Turkey for example when you use PPP(which is supposed to be an improvement). Turkey's PPP is quite high now, meaning that the Turkish lira is strong and on average Turks are supposed to be slightly richer than neighboring EU country Bulgaria. In reality, Turks are miserable because some basic stuff is taxed to extremes and the market is captured by a few groups and to achieve a Bulgarian lifestyle in Turkey you actually have to pay 2X or 3X.

Maybe we just should stop looking and these stats for a proxy to other stuff altogether. They don't work, just use whatever you are looking directly.