The NYT neglected to mention that the Congressional Budget Office has repeatedly found that adopting plans providing more choice within Medicare, like the one by Representative Paul Ryan and Senator Ron Wyden touted in this article, raise rather than lower the cost of providing care. The basic problem is that private insurers are very good at cherry picking patients -- better than government bureaucrats in preventing cherry picking. This means that private insurers will find ways to get patients who cost them less than the average payment, or less than the average risk-adjusted payment, for Medicare beneficiaries.
This is the reason that Medicare Advantage and its precedessor in the 90s, Medicare Plus Choice, raised the cost of Medicare. The Congressional Budget Office has also found that private insurers are less effective in controlling costs, which is why they projected that Representative Ryan's proposal for privatizing Medicare would increase the cost of providing Medicare equivalent policies by $34 trillion over the program's 75-year planning period.