I do get the gist but I cannot imagine that Congress has the power (or the guts) to make such distinctions which, in any event beg the question. If Obozo was a NBC (as I believe) then Congress has NO ROLE in the process. If he was NOT an NBC, then Article I, Section 8, "To establish an uniform rule of naturalization" probably confers the naturalization power on Congress and you are right.
However, SCOTUS is NOT looking for the kind of extreme controversy that would be generated if they did nothing about Obozo and interfered with Cruz declaring him ineligible. There would be busy guillotines in and on the streets on all approaches to the SCOTUS building and new aspiring Robespierres, Dantons, and their ilk until they too are separated from their heads.
The question with Cruz puts him even farther beyond NBC, as Cruz is admittedly a dual citizen at birth, born abroad.
That opens Pandora's box -- as if it wasn't opened already, by derelict Congress and chicken-scheist SCOTUS. Congress can't have the power to "make natural," both for reasons of simple logic (an Act of Congress can't change a boy into a girl, a person is born one or the other), and for reasons of preserving the nation. A rogue Congress could pass an Act that says either "all persons born in Iraq are citizens at birth of the US," or even simpler, "all persons born in Iraq shall be considered to be born in the US and subject to the jurisdiction thereof" The second form, if allowed, bypasses naturalization altogether, because the 14th amendment attaches citizenship on "born in the US" in combination with "and subject to the jurisdiction thereof." The first form looks like naturalization.
Congress always has a role - it counts the electoral ballots and has the power and duty to ascertain that the president-elect is qualified. Usually this is not contentious. But when it is, Congress has an amazing ability to clam up.
From my point of view, Congress and SCOTUS are subversive of the constitution.
I think the country is lost anyway, except as a name for a location on the planet. The people are lazy, sheepish, and gullible. I watch things unfold for entertainment.