I don't think that's the part of the sentence that's the issue - I read it the same way as the previous poster, and realise what the problem is. You said:
But in every other civilized country they wouldn't have to be able to self-assess something like that
On first read, it sounded like you're saying "they wouldn't have been able to self-assess something like that", whereas you're actually saying "they wouldn't have needed to self-assess something like that".
Please remember that this is an internet where seemingly half of the Americans we meet are 100% convinced that the US is the only place in the world with cars, electricity, running water, etc. The assertion that the US is the only country with nurses capable of self-assessing injuries totally seems like something that someone would post.