A big problem with doctor certification today is that doctors are graded on a boolean scale, pass/fail. So two physicians can both be certified, but one can be vastly better than the other, and you, as a uninformed patient, have little way of judging. A close relative of mine has had an opportunity to assist in many open-heart surgeries performed by several doctors. He tells me that there is a huge difference between surgeons. One surgeon regularly completes operations in 20 minutes with no mistakes and complications, another surgeon, at the same medical center, takes two or three times as long, is much "messier" (cutting things he shouldn't), and his patients have many more complications. It's obvious to everyone in the operating room who is the better doctor, but no one will tell patients. For patients, both doctors are "competent." In free market, presumably there would be at least one rating agency that would provide a more nuanced report on doctors.
Also, to get an MD, there is not a "uniformly high bar," at least there hasn't been in the past. Medical schools discriminate in their entry requirements based on at least race, but possibly other non-competency factors like sexual orientation and parental alumni status. So entering medical students are not held to a uniformly high bar. Once in medical school, student performance diverges. Some students are at the top of their class, others are at the bottom. Some pass the certification exams with high scores, others barely pass after two or three attempts. Most of this information is not readily accessible to the average healthcare consumer.
The primary aim of doctor certification seems to be to restrict the supply so as to keep wages high, not to help consumers choose. The best solution really is freedom, not mere "detailed technocratic tweaks" to a system designed from the bottom-up to protect the interests of the medical profession against competition.