What you've presented is not an argument against public healthcare, it's actually an argument against poorly funded public healthcare.

Conservative politicians in both of the countries you mentioned have been systematically, over the course of decades, been defunding their own government-run healthcare programs, specifically in the hopes that people like you will conclude that because of this the system is "broken" and that we must as a result switch to the private, profitable healthcare that said politicians are being paid to support.

If you want to talk about properly funded public healthcare, there are plenty of good examples in the EU to look at.