Eligibility is not a political question.
If it were there would be no need for Article II.
Courts have dismissed eligibility challenges on the grounds of political question. It’s all in how attorneys frame the issue.
For example: this lawsuit went to the U.S. Supreme Court but was denied a hearing:
Keyes, Barnett, et. al v Obama, et. al.
U.S. District Court Judge David O. Carter: “There may very well be a legitimate role for the judiciary to interpret whether the natural born citizen requirement has been satisfied in the case of a presidential candidate who has not already won the election and taken office. However, on the day that President Obama took the presidential oath and was sworn in, he became President of the United States. Any removal of him from the presidency must be accomplished through the Constitution’s mechanisms for the removal of a President, either through impeachment or the succession process set forth in the Twenty-Fifth Amendment. Plaintiffs attempt to subvert this grant of power to Congress by convincing the Court that it should disregard the constitutional procedures in place for the removal of a sitting president. The process for removal of a sitting president—REMOVAL FOR ANY REASON—is within the province of Congress, not the courts.”U.S. District Court for the Central District of California, October 29, 2009
http://ia600204.us.archive.org/1/items/gov.uscourts.cacd.435591/gov.uscourts.cacd.435591.89.0.pdf
Kerchner v. Obama, Mario Apuzzo’s lawsuit was also dismissed, in part, on political question grounds.