Answer to Question #1: No. Originalism/textualism takes the meaning as the plain meaning language at the time each Article/Amendment was written, not the late 17 century plain meaning of language for everything. So the 13th amendment, which was written in 1865 would be read in the plain meaning of 1865 where as the Articles would be read with the plain meaning used by the founding fathers. In English, it's consider increadibly rude to use the third person neutral singular pronoun to refer to a human being and the pronouns "they/them" were not a third person singular at this time (very recent development in the English Language) nor was gender neutral writing (using he/she) also a thing at the time. Because the president would always be a single human being, the writers would have to use either he or she and there are plenty of documents that have a "gender neutral he" to refer to a generic individual occupation holder when gender was unknown at this general description stage. The framers were discussing the occpuant of the office of President and that individuals government powers, so gender wasn't an issue in their mind.
Additionally, the originalist/textualist would argue that the founders would have been aware of situations where a monarch was in fact a woman. The rule of Queen Anne was monarch for 12 years (5 as queen of England, 7 as Queen of Great Britain) from 1702-1714, which was 52 years prior to the Declaration of Independence. Prior to Queen Anne, you had Queen Elizabeth I, Queen Mary I (the first truly recognized Queen of England and for whom a U.S. State was named (Maryland aka Mary's Land)). Before her was Queen/Lady Jane who ruled for all of 9 days before Mary was installed. Before Jane there was the disputed Clamaint to the throne Empress Matilda/Maude who ruled from April to November of 1141. It's important to note that with all these Queens, they were all Queen Regents (i.e. Queens who ruled, to be contrasted with Queen Consorts, which were Queens whose husbands ruled. In English, a male equivelent to a Queen Consort is called a Prince Consort. There is a fun historical example of this difference mattering in a title in Polands King Jadwiga (1384-1399) who was in fact, a woman. The use of the term King as her title is due to the fact that the Polish Language the word "King" is gender neutral and always refers to a ruling Monarch while the Polish word for for "Queen" always refers to a female Consort of the Ruler. While English word Queen is acceptable in this situation, it's insulting when translated back into Polish. Though it should be pointed out, at the time, Polish nobility didn't want to use the female title cause they couldn't stand her betrothed fionce, Duke William of Austria, who would have inherited the title "King" had he married "Queen" Jadwiga. By crowning Jadwiga "King of Poland," William would only get the title of Prince Consort. For some odd, William decided to not marry Jadwiga upon learning this and getting kicked out of the Polish Court.). Additionally, contemporary to the declaration of independence's signing was Tsar Catherine II of Russia, better known as Tsar Catherine the Great who also took some small naval actions in the aid of the Colonies at during the Revolutionary War. Because the founders would have known about this and did take actions that would have barred a situation similar to Catherine The Great from happening in the US (I.E. Catherine was not Russian but a German princess who married the Tsar that preceeded her reigne. The Constitution Explicitly states that the President must be a citizen of the U.S. by birthright) had they wanted to exclude women, a similar constraint would have explicitly been said. While the OP's quote comes from Article II Clause 1, the qualifications for the officer are taken from Article II Clause 5 (emphasise added):
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.
The use of "No Person", "Any Person" show that that not only did a potential president not have any explicit conditions on their individual gender, but the founders went out of their way to eliminate any indication in the qualifications clause whatsoever. Had they wanted it in there, they would have said as much (consider that elsewhere in the original Articles, they explicitly dehumanize slaves, we can assume they weren't bound by anything we'd remotely consider "PC" language.). The use of the word "Person" is especially important as Textualists/Originalists veiw the use of the word "person" or "people" in the Original Constitution (as drafted) and the Bill of Rights to explicitly mean "an individual citizen" and usually was gender neutral (This is muddled in the second amendment as at the time, Militia is contextually read to mean all able bodied men of a community (though down to the bare minimum of one able body men if that's all the community had) though through the same lens of the textualist/originalist, the part about the militia is a justification for protecting the "right of the people" (read: right of an individual) against government intrusion. As we'll discuss later, the founding fathers did make an amendment stating that they were too lazy to write down ALL the rights of the people.
All of this is before the 14th amendment, which garentees that all citizens are granted equally and all subordinate governments must not violate the Federal Constitution. Additionally, we have the 9th Amendment which says that just because the constitution does not explicitly list something as a right of the people, does not mean that right can be deniegned to the people. The 10th Amendment takes the 9th Amendment and denigns the Federal government the power to legislate on anything that denigns a 9th Amendment right to the state or the people (In effect, it's a nice way of the founders saying: We're lazy and don't want to go through every possible situation, so when you're in doubt, assume it's a right of the state, unless they don't cover it, then it's the right of the people). The 14th Amendment effectively patches up the loop hole created by the 9th and 10th amendment (i.e. while the Federal Government couldn't denign people certain rights, the State Governments were not bound to these restrictions and they could and did use this loophole in many ways, most notably Slavery).
It should be pointed out, a lack of Universal Sufferage was not prescribed by the constitution. At time of drafting the only direct elections in the Federal Level were for House of Representative members, and the Constitution said that in order to be eligible to vote for Represetatives at the Federal Level, you must be eligble to vote for State Legislature representatives. Voter eligibility was deligated to the states to decide (Many states did grant women the vote prior to the ratification of the 19th Amendment. Just not all of them.). Unamended, the Constitution delegated election of senators to the State Legislatures and the President was through electoral college, and the states determined how electors are allocated (though almost all pick by the candidates state wide popular vote. The two that don't go by district popular vote with the overall statewide popular vote winner getting two additional votes).
Now, with all that out of the way, to answer question 2, the Living Constitution might actually agree with your assesment that using a third person singular masculine pronoun (He) shows that under modern language rules, the Constitution limits presidential eligibility to only men, but they would also have to ignore all of Article II Clause V where eligibility is outlined in gender neutral language. Living Constitutionalists do not ignore sections of the constitution, but rather read the language in modern plain language (thus rules change as the common definitions change). There are circumstances where orignilalists/textualists AND Originalists would read it the same way (though they might see the other side's read as "Right for the Wrong Reasons").
One final note, but the difference between Originalists and Textualists is the acceptance of outside commentary by the Founding Fathers in interpretation. An Originalist will also cite examples from history that the founders might have been aware of(ala my listing of various Queen Regents), to their contemporanious writings (Hence why you hear a lot about the Federalist Papers and the Anti-Federalist papers as both explained the reasoning be hind the Unamended Constitution and the Bill of Rights respectively) and even drafts of articles and amendments that did not make the final cut. Textualists will concern themselves with the plain meaning of the language used but would not reach to outside sources to justify what the founding fathers meant when they wrote it. If they had meant something different, they would have written something different.
As an example, an Originalist might side with someone on the right to a religious exception to the Constitution because when drafting the Second Amendment, there was originally a clause about respecting religious objections to bearing arms in a Milita. The Textualist would come to the same conclusion, because the First Amendment restricts the government from unduly burdening the exercise of their own personal religious beliefs (the Originalist would agree on this point too, but the draft about this as well, as the line in the second amendment was cut when the first amendment made it redundant). Here, a living Constitutionalist would agree with the Textualist but not the Originalists.
Textualists ask "What was the writer saying?"
Originalists ask "What was the writer thinking?"
Living Contstitutionalists ask "What are we hearing?"