Just to clarify: you do NOT need to be a citizen of an EU country to have the EHIC card. You only need be a resident.
That said: I had the misfortune of having to use my card last week and it was worthless in my situation. I broke my collarbone during a ski holiday in Italy last Wednesday. I had to go to a private doctor, who would NOT accept the EHIC card. I had to pay him out-of-pocket (230 Euros including emergency room visit, xrays and drugs). When I returned to London, I tried to get reimbursed through the EHIC but was told I needed to have picked up certain Italian reimbursement documents that could only be obtained IN PERSON from the local health office in Italy. And even if I had the documents, I was told I would only get back 50%. (The EHIC card gives you free or reduced health care. Obviously in Italy, it's only reduced care).
So: I have now submitted my paperwork to my travel insurance company, Insure And Go. Stay tuned!