It's actually easier to shop for medical care than dental. Health plans are now required to give members an online price comparison tool. You can just log on to your insurance company's member portal and search for the care you need to see prices for network providers in your area. Unfortunately, the same rule doesn't apply to dental insurance.
Still, a problem at least as big as finding various options and their prices seems to be finding somebody incentivized to give an honest evaluation of whether a given treatment is medically necessary at all.
Sometimes I wonder if it wouldn't make sense to have a doctor "on a retainer" that just gets a fixed yearly compensation for advising what to do, and what to better decline, but there's probably tons of ways this could go wrong in either direction (overtreatment vs. missing important issues) as well.
You are essentially describing "concierge medicine". This can be a good option for affluent patients who can afford the monthly fee but it's not a scalable solution for the systemic problems in the US healthcare system. A lot of the people who take advantage of it are what is known in the industry as the "worried well" — rich hypochondriacs willing to pay for personal reassurance whenever they have a tummy ache.
https://www.cms.gov/healthplan-price-transparency/consumers