How was George Romney eligible to run for the presidency?

[True Federalist (진정한 연방 주의자)]

It was raised, but his parents had kept their US citizenship while they were living in Mexico, and they never gained Mexican citizenship, so George was a US citizen from birth and never was a Mexican citizen. (Mexican law at the time defined citizenship by jus sanguinis alone.)  Whether he qualified as a "natural born citizen" depends on how one defines that phase.

The definition included in the Naturalization Act of 1790, was not included in its successor, the Naturalization Act of 1795 or any law since.  Arguably, the 14th Amendment limits natural born citizenship to jus soli alone, in which case neither George Romney nor John McCain would have been eligible.  Whether jus sanguinis alone makes one a natural born citizen or not is unsettled, but probably would be, especially in a case like George's, wherein he never had any other citizenship despite being born in Mexico.

[True Federalist (진정한 연방 주의자)]

Why does it seem to me like presidential eligibility rules are such a gray area?

Because the gray areas affect so few people, and never to someone who won, so that it has never really mattered.  However, John McCain's claim to being a natural born citizen derives from jus sanguinis alone, the same as George Romney.  Even if John was born in the Canal Zone (a matter of minor dispute), it was at a time when jus soli did not apply there, but was retroactively applied.

[True Federalist (진정한 연방 주의자)]

Get your dates right. The relevant acts are mentioned above.  Jus sanguinis was part of the explicit law of the land from 1790 to 1795.  Both acts were passed by the Federalists, as was another law passed in 1798.  It was the 1798 act the Republicans repealed in 1802, thus returning to the 1795 law with a few minor changes.

[True Federalist (진정한 연방 주의자)]

Well if his father wasn't a US citizen, that does make a precedent to the question as to whether someone who doesn't have a citizen parent at birth is a natural born citizen, not that was ever seriously questioned by anyone besides the standard loons...

Heck, there were some 19th century birthers who asserted that Arthur had been born in Ireland, but they dialed back their claims when no one believed them.  It's unclear if Arthur's dad was a US citizen at Arthur's birth.  It only required 5 years residence to take citizenship back then, so he could have been a citizen by the time of Chester's birth.

[True Federalist (진정한 연방 주의자)]

To be fair, the natural-born citizen clause itself makes no sense. In every country, but particularly in a country like USA.

     Indeed, it is quite ironic that a country of immigrants precludes immigrants from its highest office. I've heard that it was due to a fear of royalists taking power, but I don't know how much truth there is to that.

That was indeed one reason.  That's also why the clauses prohibiting the granting of titles of nobility by either the States or the Federal government were included.  There is also an amendment that was sent to the states in 1810, but not yet ratified that would have stripped U.S. citizenship from anyone accepting a foreign title of nobility. (Jerome Bonaparte had married a Baltimore gal, and with the situation with Napoleon, there was some worry that Napoleon would give their kid a title and someday use him to interfere in American politics.) It got the approval of 12 states at a time at which only 13 would have been sufficient to adopt, and there were some who thought it did get adopted as our Thirteenth Amendment because of confusion over whether South Carolina or Virginia had adopted it. (In both states only one house approved.)  However, once the War of 1812 began, anti-French sentiment was muted and after it ended France was no longer a threat, so the number of States that have ratified has remained at 12 since 1812.  It would take 26 more States to ratify it today.

[True Federalist (진정한 연방 주의자)]

Actually, question for Ernest:

Birthright citizenship becomes the law of the land in 1868 through the 14th.  However, birthright citizenship was made retroactive.

Would a non-citizen born on US soil in, say, 1840, given citizenship by the 14th, be a "natural born citizen" eligible to run for President?

That depends.  I take a fairly narrow view of the "and subject to the jurisdiction thereof" qualifier.  In the case of a former slave or free negro, yes.  In the case of an "Indian not taxed", no.