Without looking too deeply into the Medicare regs, a few things may be worth noting.
1. The article linked implies that hospital records of the birth are a sufficient "documentation" of the infant's citizen status.
2. For most Medicare benefits, an evidence of "need" has always been a requirement. The mother's need should already have been established, or she wouldn't have received Medicare assistance with the birth. This means the hospital should have all the information needed to make the application for the infant.
3. The requirement to confirm eligibility of the child does not prevent a hospital from providing treatment. It just means that if "eligibility" isn't confirmed they won't get paid (by Medicare) for treatment rendered. Most hospitals arent' too stupid to be quite sure of what's "approvable."
4. In all cases I've seen, payment of Medicare benefits is retroactive to the time at which a patient becomes eligible for coverage, regardless of how much later the determination of eligibility is fully documented.
5. Medicare in most states generally is so slow at paying that a month, or a few months, delay to get the paperwork done before payment should be routine for any medical practice that deals with Medicare patients.
(6. Private insurers often aren't much better in the "prompt payment" area.)
There may be real substance to this report, but quite a bit of additional digging would be necessary to really tell if the reported "change" actually means much. As the article exudes a faint aroma of "politics" I'm not sure how much digging I'd like to do; but changes in Medicare regs and policies are of sufficient personal interest that I suspect I'll attempt some follow up. It may take quite a while.