The American medical insurance system is totally fine with quackery. Most plans also cover acupuncture and all sorts of other nonsense as well. They are not overly concerned with efficacy. They are concerned with what will attract new premium-payers, and what is cheap to cover on the back end. Quackery is great this way, because it’s generally inexpensive and a lot of people want it.
As for doctors, I’ve never personally heard of an evidence-based medicine doctor referring to a chiropractor unless the chiropractor also does other things that are evidence-based, such as massage therapy. That said, I won’t say it never happens because the other factor here is lobbying. The chiropractic industry has engaged in an aggressive decades-long lobbying campaign to legitimize themselves. They have managed to get laws requiring coverage of their services, they’ve manage to get licensing systems in place, they’ve generated reams of garbage science, etc. It’s one of the oldest and most organized bits of quackery, so they’ve come a long way.
It was born in quackery and has remained there ever since D.D. Palmer made it up in Iowa. The founding principle of the practice is that god sends life energy down through the top of your head into your spine, and all health ailments are caused by “subluxations” interrupting that energy. A subluxation is claimed to be a misaligned vertebrae, but of course if you had that you would be dead.
Nothing chiropractors do is evidence-based. Many of them are also licensed massage therapists or do other evidence-based things, but nothing that is unique to chiropractors has any value whatsoever.
It’s extremely common in North America also. That doesn’t make it effective. People do all kinds of things that they think are making them feel better but are actually doing nothing. It’s not a criticism of the people who use these services. It’s why we need medical regulations to prevent quackery. We can’t all as individuals be expected to suss out every scam and fake medical practice. Lots of alternative medicine seems pretty good if people don’t know all the science or lack thereof.
Chiropractic has no evidence to support what it does, does nothing at best, is mildly dangerous at worst, and shouldn’t exist as a practice.