"Cash" here means 'not wacky insurance schemes'. Their FAQ says

"To keep our prices as low as possible, cashier’s checks or cash are the methods preferred. Credit cards are accepted on a case by case basis. Human resource departments or divisions of self-insured companies can make other arrangements if necessary."

(They also have a partnership with some sort of personal loan/financing company which presumably would take credit cards or bank transfers.)

It would be silly for a doctors office to accept only suitcases full of $100 bills, not only would that be highly impractical for both parties, it would make them a prime target for robbery so they'd have to invest a ton in physical security.

"Cash only" means "not insurance claims" in this context - usually when you go to the doctor you give them your insurance information and your expected copay the doctor (actually their admin staff) then submits a claim to your insurance company for payment. If the company doesn't pay, or pays less than expected, they will bill you.

Large "cash" transactions aren't a problem.

Multi thousands of dollars+ "cash" transactions are done all the time (such as a down payment on a house), you go to the bank and get a cashier's check. There's no reason you couldn't take out cash instead of getting a cashier's check, but it wouldn't be secure or practical. And your payee may have an issue with accepting physical cash from a risk perspective.

"Paying cash" in this context doesn't mean a briefcase full of $100 bills. It means direct payment from the consumer to the provider. That direct payment could be in the form of a check or credit card.

Pretty sure they can take a debit card, which even in the U.S. is a chip + PIN transaction, with a minimal transaction fee rather than the substantial percentage applied to credit card transactions. Anyway, by cash they mean full payment now, rather than invoiced to an insurance company.

You should maybe read the article then since it's about something else.