My takes, which aren't worth much around here:
He said he was told he might be able to, didn't say he was going to do it. Heard the interview on the radio, not sm
Our President seems like the very epitome of guy-in-charge-who-wants-a-yes-man, with plenty of evidence to support that belief. Therefore likely asks questions until he gets the answer he is demanding.
And why were they here? Lawfully immigrated and processed? Registered as aliens? Or just passing through? Just taking advantage of a situation? Why does the first section say "born AND subject to the jurisdiction..." - obviously not everyone born here.
Most reasonable reading of that, to me, is that they aren't part of an exception, most obviously being that of foreign diplomats/ambassadors/etc. For example, Sergey Kislyak in the US as ambassador to Russia, is here directly for, on behalf of, and under the jurisdiction of Russia. In fact, he would fight being subjugated to any control of the US.
On the other hand, an immigrant, legal or otherwise, likely entering the country purposefully to avail themselves of the US' jurisdiction, not to fight it. Joe German-Name doesn't move to little Minnesota and declare the land for the fatherland and expel all others, he subjects himself to taxes, policing, and social policies; i.e. subject to the jurisdiction of the US.
Otherwise, who specifically would be born here and subject to the jurisdiction, that isn't already going to be considered a citizen? That would alter far more than just illegal immigrants.