Both of the links you provide corroborate what I said. Mass-market media tends to do a terrible job with reporting nuanced details like the medical billing process (or computer security), but even this watered-down CNN overview notes:

> The differences can be stark. Private insurers allow an average of $1,226 for low-back disc surgery, while Medicare will only permit $654, for instance. And the gap can grow wider depending on where the patient is. In New York, insurers allow $1,352 for a gall bladder removal, compared to $580 for Medicare. Some services are more comparable. For office visits by established patients, for instance, Medicare will allow 92% of what insurers do.

As for Table 1 in the paper you linked, every single city listed there receives less from Medicare than the black-box FAIR estimates. In the case of San Francisco, Medicare basically pays 33% of what the private insurers are estimated to pay.

And that's just tracking Level 3 office visits, which is where the Medicare shortfall is generally the smallest (generalized and routine care).