So if a 30 year old person is elected and sworn in, a person who is not eligible, and removal can be effected only by impeachment, what is the purpose of Article II?
Eligibility is a legal matter. Legal matters are handled judicially.
If the courts rule that a candidate is ineligible, it is still up to the election authorities to remove that candidate from the ballot. This happens quite frequently. It is not automatic. The courts must have a case before them rule upon it, and then the proper electoral authorities must act.
However, if an ineligible candidate is not removed from the ballot and elected, no successful case having been brought against him, it is then within the purview of the body to which he was elected to remove him should they decide that later evidence is valid. In the case of an ineligible candidate, such as a governor, achieving the office and later ruled by the courts to have been ineligible, it is not automatic. Tthe state legislature must vote to remove him. The courts themselves do not. This has happened.
In the case of the POTUS, the courts from the lowest to the SCOTUS have decided either not to hear the case on its merits, rejecting it and subsequent appeals on various grounds such as jurisdictional issues or lack of standing, or have otherwise dodged the issue, which they have admitted.
Should the SCOTUS decide to hear an appeal on the issue of Obama's eligibility and define Article II, that ruling, if against him, would still leave the matter of his removal to Congress. It is not automatic.
The system failed in regard to Obama while he was a candidate. That was the time to get rid of him. The legacy of this failure is obvious: candidates and their supporters now seem to feel that they can claim eligibility by proclaiming it to be so, without benefit of judicial review. Unless cases are brought before the courts, and heard, the concepts of constitutional eligibility for office ... even the concept of citizenship ... become increasingly irrelevant.
The purpose of the age provision is to limit the pool of candidates from which the electors select presidents to persons who are 30 years old or older. Nowadays, voters select the electors in all of our states and so voters should also confine their choices to persons who meet the age limit.
Eligibility is a legal matter. Legal matters are handled judicially.
If you try to flesh out these assertions, i think you'll find that any question under the sun can be called a "legal" issue if you type up the question, put it into a manila case file and set it on a judge's desk for his resolution. To just declare that something is a "legal matter" to be "handled judicially" is to take an inappropriate shortcut.
Our Constitution distributes and entrusts power and authority to various constitutional actors to perform various decision-making functions. The Constitution empowers the electors to select presidents. There are no provisions calling for judicial review of their decisions.
For some reason, some people seem to think that only judges can be trusted to interpret and apply the eligibility standards for presidents as if it were some sort of mathematics problem that can only be resolved by a board of mathematicians. What makes you think that Supreme Court justices would be unanimous in choosing the same precise definition of the NBC clause?
Maybe it would help you find some peace here if you attempt to come to terms with why the Supreme Court has not become involved in the questions regarding Obama's eligibility, why the Supreme Court has never even hinted that it has to power to disqualify presidential candidates, why the justices attended both of Obama's inaugural ceremonies and why the Chief Justice has on two separate occasions volunteered to administer the oath of office to Obama? In view of all that, I just don't see how anyone can conclude that the Supreme Court (1) knows that eligibility is a "legal matter" that can only be "handled judicially," (2) knows that Obama is not eligible to be president, and (3) that Obama is, therefore, not really the president.
And, if after all that, you conclude that the Supreme Court obviously does not agree with all of your analysis, then how are you going to avoid taking that shortcut of yours to assert that the proper scope of Supreme Court power must be just another "legal matter" that can only be "handled judicially" by the Supreme Court - and not you? Can you accept the Supreme Court's determination that it lacks power to overrule electors?
I think it's obvious from the text of the Constitution that the electors have been entrusted to apply the eligibility standards and to select our presidents. I don't see any provisions for a role by the Supreme Court in selecting our presidents.