I'd vote pretty much any of (off the top of my head): Erastil, Sarenrae, Shelyn, Arshea, Andoletta, Lymnieris, Desna, Abadar, Tanagaar, Kroina, Duellona, Curnunos, Korada, or Apsu. There are likely others, but those come to mind.
Some of the Empyreal Lords are "weak" votes, comparatively, but I'd hands-down vote for them above anyone on the ticket we've got now...
If we're talking more generic outsiders, hands-down an empyrean, followed by a solar. While I prefer lawful good over-all, an empyrean has the best balance, mentally and a solar second-best - a veranallia is more charismatic, but nowhere near as intelligent, and unfortunately there is just no archon that has been published that measures up to the standards of the other celestial and infernal hosts (star archon at CR 19 being closest, and it is undersold, compared to a solar, mentally, in all categories if only by a narrow margin). Second to solar would be a draconal, actually, then planetar, actually, then star archon, then veranallia, and then gate archon.
Now, all that said, I'm mostly following the "REAL Outsiders" concept to the letter and ignoring the actual laws of our land... though it could be argued that all of them were "born" here due to their original conception and development and printing here in the United States, I could see that being a dubious argument if we're presupposing that Golarion actually exists and the laws of the land of the U.S. do as well (which Golarion actually presumes). So!
... what kind of "REAL Outsider" could be born here in the U.S. (at least hypothetically)? Though it would have required immigration of its parents from its native lands, a coualt could, presupposing Earth religious myth-as-inspiration-for-Golarion-reality, have been born here in the actual United States of America. Their lifespan is indicated to easily be long enough that they could pass the minimum required time with no problem. They are extremely intelligent, wise, and charismatic, and have excellent diplomatic and sense motive skills - ideal traits in a leader.
If that doesn't satisfy for some reason (because it looks like a snake, maybe? Not human enough? I dunno) I'll probably have to switch gears. Maybe this "Endranni Malesk" that I know very little about other than a) he's an aasimar (so, technically a "REAL Outsider") who is lawful good (yay!) and CR 14 (solid) with at least some bard (social!) and cavalier (huzzah!) levels. Similarly, possibly an NPC named "Koss" - apparently a CG (yay, good! ... though less, 'cause Chaos, it's still good!) azata-blooded sorcerer (awesome!) with a CR of 10.
Those are the highest CR aasimars I can find on the chart.
The problem, of course, is that those aren't born in the U.S.A. - they were born elsewhere, which disqualifies them from the office of Presidency.
If we were looking for a creature who would hypothetically (if not practically, given our actual real life) be born in the U.S., and disqualify them based on not being "human enough" then I'd probably have to go with something like a half celestial nymph (preferably with the big boosts pumped into mental attributes, though constitution would be good, and much more charisma isn't that necessary; similarly, since we're talking templates, I'd squeeze a couple more in there if I could, and probably rearrange some skills, but all of that isn't, strictly speaking, necessary - only the half-celestial template is to guarantee/reinforce her goodness and to make her a "REAL Outsider" as per the title).
But that's kind of a, "Here's my (sort of) ideal candidate for president, if I could just kind of off-the-top-of-my-head grab from creatures to vote for." kind of thing.
None of that really answers the OP.
How would it be different? It couldn't really be expressed.
The first thing is that said Runelord wouldn't run - he'd conquer. If he deigned to run according to the law instead of merely running roughshod over our puny forces, he would simply have all would-be opponents executed by "non-affiliated entities" that he would publicly denounce and probably lead the hunt for and execute in a very obvious manner (which would, of course, be staged - his loyal servants wouldn't be dead for real, but fake ones would be; Illusion is tricky, you see).
Generally speaking, opening it up to outsiders of second generation (or those who could fake it) generally means you're leaving only the bad guys*. Congratulations: I didn't know it was possible, but you've somehow made this election worse**.
In the end, whatever has the ability to get triggered suggestions implanted (and the highest charisma check, if multiple creatures do), and highest relative mobility (to place those triggered suggestions into all the voters) will win. The suggestion, of course, is "vote for <chosen candidate>" and the "trigger" is "when you look at the vote booth" - the only trick is getting the people to vote in the first place. The best bet, then, would be to set up loyal suggestion-placers at the voting-places (probably hidden).
Of course that would be deemed blatantly illegal. Trivial for a master of Illusion to act out, however (what with invisibility and simulacrum being on his spell list). Of course, something like a Lawful Good gold dragon who was trying to oppose such a tactic could engage in the same ploy.
* Not, technically, true. Anyone who doesn't respect a local land's laws would, of course, go forth with their plans; generally (though admittedly not always), however, it is malevolent creatures tend to do so in order to gain temporal power...
** Just kidding. That's not possible.