The limits in many European systems appear to be on self-determination. There are real barriers in some circumstances to getting second and third and fourth opinions, additional tests, treatment for infants born premature before a certain week, novel and unproven procedures, etc.
I don't know which healthcare model is preferable. It appears to me the US and European models grew under different historical circumstances and have different failure modes.
Again, not trying to spark debate. Just contributing a personal perspective having lived on both sides.
I'm OK with it too (in fact have used it since my previous employer offered insurance, so why not) as long as people understand that it's a convenience for people for whom time is more important than money and who want to get prompt treatment for things that are important enough to stop you working but not enough to kill you.
Having heard all of the horror stories from the US I was quite worried until he eventually told me how much I had to pay extra - pretty sure it was £4.20. :-)
Best of luck to you and for your kid.
I can only speak for Switzerland, but an educated guess is that this is the default just in about every European health insurance system.
And no, you can't be denied (basic) insurance (which covers everything necessary) for a pre-existing condition.