hell no, the NHS is just about the worst, that's where the years waiting lists for emergency operations and shit came from. thats because its totally defunded. canada is okay with a lot of funding but it could be better too.
full public with private options is definitely the ideal in my mind. for example i could do certain mental health related things here but its a very slow inadequate process here when i used to get private care for like 20-30€ for prescription appointment and free checkins related to that from that point on. i feel like a subsidized private care acts as permanent accesible cheap care that with any decentness, will keep the freemarket from price gouging and engaging in cartel tactics as much. they'd actually have to be better or specialized to get business not just be all shit.
with illegals definitely a problem and needs to be dealth with harshly, but if talking of the EU in general, americas got something like what 11-12 million illegals? eus got 27 million non-eu citizens here. the numbers on illegal specifically are somewhat hard to find, but in 2015 during peak migrant crisis 1.3mil came in. i should add, while they may get emergency treatment, generally you have to apply for social health insurance if you're not a citizen, even if your from a different eu country, and for cross country workers this can sometimes make private care quite appealing. i think america would benefit from not being quite so generous with social health insurance (as would certain more liberal western european countries)