There are plenty of folks who travel to specialists in countries with Universal Health care. Many sports pros go to specialists in different European countries depending on their problem. (They also go to specialists in the USA).
We have some good friends who traveled to Nantes, France for surgery. She had a condition called
Pudenal Nerve Entrapment for which surgery was available in Houston but was not covered by their HI company at the time as it was still classed as experimental. (They live in Louisiana). After trips to Houston for tests to confirm the diagnosis they faced a $25k bill for surgery or they could could go to Nantes where they could have it done privately by the surgeon who pioneered this particular surgery. The total cost of the surgery and hospital stay was <$5k so they both went to Nantes and spent about 10 days there. It still took awhile for recovery but she is doing great and they came to stay with us in England for 2 weeks last year and they did lots of driving on their own, plus some walking trips with us.
The UK, in particular London, is a popular destination for patients seeking treatment. In the past I've read of a US HI company on the East Coast offering to send heart patients to London for bypass surgery, paying for a 2nd person to travel with them including the cost of several nights stay. I think it was Guy's Hospital, that has a world re-nowned Heart center, that was the destination.
Medical Tourism