If there's not a table somewhere maintained by the US government that associates social security number with citizenship status, that's because a choice was made by the government not to do that. It would be a simple enough thing to do.

(yes, checking against name / DOB / ssn always has some inherent messiness to it)

It's definitely a choice, because we've avoided having a real standardized identity system run by the government for so long.

But there are reasons for people to oppose it on both sides of the aisle (states rights, immigration views, anti federalism, libertarians) so it's a pretty hard task. Maybe this admin could try it as an immigration security measure and get some support that way but I have my doubts.