I don't know what the heck to call it. "To Chicago" is definitely a simple prepositional phrase. I think of "on" in these sentences as a shortening of "onward," which can actually stand alone as a command, or instruction, or direction. It's like "Forward ho the wagons!" "March!" "Charge!" "Halt!"
"Continue on to X" is fine. (I used it in my first post, as did you in your first post.)
But I meant to add that the "to" goes with the following noun to make a prepositional phrase. It doesn't go with the "on." So "continue on to" would not be the expression. A preposition may not be required. "as we continued on skyward." "Continue on westward." (imperative)
In this "continue on" group, the "on" might be considered optional, which makes it a bit different from Kennedy's usage, even though the "continuity of travel" is implied.