Elective or cosmetic surgery isn't necessarily a natural monopoly: unlike, say, emergency care where you tend to go wherever the ambulance takes you, or GP care where quality is difficult to judge without spending years with a particular service, or even insurance where you suddenly find out that the service sucks when your life, or at least your quality of life and your employability, is on the line and they drop you or screw up your coverage. People getting cosmetic and elective procedures have plenty of leisure time to shop around, and because they aren't things that you need to have to live, they do marketing and directly compete on price. It's part of why lots of countries with nationalized care (single-payer or otherwise) make exceptions for those services and they are covered by out-of-pocket, private insurance, or maybe only a partial subsidy.
Eye care (and sometimes dental) are often caught in the middle ground between true electives and the core system. I wish they were better covered here in Canada, because they seem necessary enough to me, but I recognize that not everyone agrees with that—and that in both cases there are lots of tricky aspects that shade between necessary and elective, like when your orthodontist is trying to upsell you to “improve” your jawline when really you just want your gums to the healthy.