Do medical clinics decide their own prices?

Sure they could

Good luck finding Drs, nurses etc who would work for the low wages you would pay due to low revenue 

"I was curious if someone could technically open a cheap medical clinic that does not take insurance..."

Yes. Physicians are free to practice as they wish. They are not required to accept insurance. There are "concierge" and subscription-based clinics that charge members a flat rate or have what is essentially "ala carte" pricing for services. These clinics build up memberships or clientele pools and do not bill insurance.

"...and only charges reasonable prices to stop giving the healthcare field so much money and power"

There's the trade off. These clinics are not cheap. Why is that? 

Well, they have to buy their equipment and supplies from the same places as other clinics. Since they are smaller, private entities they don't get the bulk discounts for buying in large volumes. Need an MRI? Well, they either have to contract with a local radiology clinic or buy one on their own and deal with all of the costs of running it. In either case, that costs a lot more than if you are part of a large health system that has radiology services you can refer your clients to. 

Need labs done? Well, machines and reagents aren't free. They have to buy them from the same lab supply companies as the big health systems. And once again, they're small so no big discount on volume. If you want to send your client out to a lab they're either going to have to pay out of pocket for it or you have to work out an arrangement with the lab to refer your clients at a discount. And AGAIN you don't refer thousands of patients from your tiny clinic so there is no incentive for them to give you a price break. 

Need to go to the hospital? Have surgery? Your doctor needs to have admitting privileges. A hospital doesn't have to grant privileges to anyone they don't want to. If you're not part of their health system, they don't have to grant those privileges. If you're not part of ANY system because you're a small, cheap, private clinic? Good luck. You can nickel and dime yourself to death as a private practitioner trying to run your own clinic outside of a system. You're responsible for all of the overhead, hiring, management, billing, etc., etc., etc. Doctors aren't part of insurance plans and health systems because they like them. It's because operating outside of them is very expensive and time-consuming. Some docs have indeed found their way around this and only deal with private-pay clients. But they are far from being able to "only charge reasonable prices" because they pay the same astronomical prices for everything as the rest of the health world does. 

Not like you think. Overall, prices are linked by the feds to billing codes-that is the set amount you can charge for a level 1-5 visit (and they determine what a level visit is) as well as codes and prices for everything else-from xrays to point of care tests, to splinting, ace wraps etc.  If you accept medicare, or bill insurance etc you must use these codes. If you want to just be fee for cash with no billing what so ever, you can set your own, which is what low cost & free clinics already do, but you can not bill medicare, medicaid or insurances. You must be aware, however, the cost of operating such a clinic you would need to cover all expenses hard and soft, including malpractice, supplies, things like site licenses, lab, xray, personal, medications, instruments, personal expenses, charting software, taxes, etc etc etc. Its not cheap. A box of suture material can be $300 for example. At out clinic a routine visit (level3-4) is $180 which is actually pretty cheap, all things considered.

I don't know. But if you open that clinic let me know. I'll be glad to let you take over my health care. You might have a regulation or two to worry about, and might need some simple things like qualified staff, doctors and such, but it should be easy. I'd love the get the high tech top notch medical care I'm accustomed to at a cheap clinic.

Could someone, yes.  But how many customers do you think you will get if you don't take insurance/