Under most circumstances, those benefits take time to kick in. In Ontario, for example:

http://www.health.gov.on.ca/english/public/pub/ohip/wait.htm...

Socialized health care is not the same as having money to pay for treatment. Some Canadians go to the US to get treated for cancer.

In Denmark there is also government healthcare, but some people, including politicians, still pay private hospitals for treatment.

Canadians go to the US for treatment because they either want to get treated faster than it is possible in Canada or they want or need a method of treatment which is not available in Canada. (The latter is not terribly surprising if you consider population sizes.) Nobody would go for US treatment because of money issues.