constitutional questment ii: eligibility for the presidency

I was thinking about who ca be president -- the eligibility rules and whether Congress can change them by simply passing a law.

Backing up, let's start with the eligibility requirements, as defined in the Constitution. Article II, Section 1 reads as follows: 

The executive Power shall be vested in a President of the United States of America. He shall hold his Office during the Term of four Years, and, together with the Vice President, chosen for the same Term, be elected, as follows: 

Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector. 

The Electors shall meet in their respective States, and vote by Ballot for two Persons, of whom one at least shall not be an Inhabitant of the same State with themselves. And they shall make a List of all the Persons voted for, and of the Number of Votes for each; which List they shall sign and certify, and transmit sealed to the Seat of the Government of the United States, directed to the President of the Senate. The President of the Senate shall, in the Presence of the Senate and House of Representatives, open all the Certificates, and the Votes shall then be counted. The Person having the greatest Number of Votes shall be the President, if such Number be a Majority of the whole Number of Electors appointed; and if there be more than one who have such Majority, and have an equal Number of Votes, then the House of Representatives shall immediately choose by Ballot one of them for President; and if no Person have a Majority, then from the five highest on the List the said House shall in like Manner choose the President. But in choosing the President, the Votes shall be taken by States, the Representatives from each State having one Vote; a quorum for this Purpose shall consist of a Member or Members from two thirds of the States, and a Majority of all the States shall be necessary to a Choice. In every Case, after the Choice of the President, the Person having the greatest Number of Votes of the Electors shall be the Vice President. But if there should remain two or more who have equal Votes, the Senate shall choose from them by Ballot the Vice-President.  

The Congress may determine the Time of choosing the Electors, and the Day on which they shall give their Votes; which Day shall be the same throughout the United States.  

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.  

In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected.  

The President shall, at stated Times, receive for his Services, a Compensation, which shall neither be increased nor diminished during the Period for which he shall have been elected, and he shall not receive within that Period any other Emolument from the United States, or any of them.  

Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:-"I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States."

The fifth paragraph above defines three eligibility requirements: natural born citizen, 35- years old, a resident for 14 years. Easy, peasy, George loves Weezie.

So, of course, Congress cannot loosen the requirements. Suppose passed a law saying that the minimum age requirement is 30, or that you don't have to be a natural born citizen. Such a law wouldn't pass constitutional muster.

But suppose Congress tightened the restrictions. For example, suppose they passed a law saying simply "No one shall become President who shall not have attained the age of 40." What then? Such a law wouldn't contradict the Constitution. The document doesn't actually prohibit the addition of more requirements.

So, based on that, it would seem that Congress could add requirements. But I suspect that that would rankle some, and would be met with the argument that that was not the intention. I'm not convinced by such an argument.

But the Article above also defines the election process*. What if the electors choose a 35-year-old? The law passed by Congress prohibits that person from being President. But if he or she was elected by the process defined by the Constitution, then I don't see how the law passed by Congress can actually override that election.

So I guess I arrive at the same place as most people, but I take a more circuitous route.

*The process was changed by the Twelfth Amendment, but not in a way that materially affects this analysis