Impeachment is a POLITICAL process, LEGAL matters are Judicial.
Impeachment is a political act of the Legislative Branch.
Conversely eligibility is a legal matter determined Judicially.
If eligibility were a political matter then there would be no set standard and the mandate of Article II § 1, cl 5 would become an arbitrary and shifting standard – a party controlling the Legislature & Executive could with impunity protect an ineligible person in the Executive. Article II § 1, cl 5 would be surplusage.
Article II commands that an ineligible person shall not be President. A person who is not President can not be impeached.
Article II does not distinguish between ineligibility prior to an election or after, a person failing to meet the requirements is at all times legally disqualified.
Nothing can be added to the text, it must be taken as it is. If the Framers intended an exception they would have written one.
“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.” U.S. Const. art. II, § 1, cl 5.
A Judicial determination of eligibility is proper, indeed it is solely for the Judiciary to determine and for no other Branch.
The Legislature has no power to interfere with a Judicial determination of ineligibility.
There is no role for the Legislature, to allow an ineligible person to remain in Office on the premise that the Legislature has not impeached would obstruct the purpose of Article II § 1, cl 5.
A Judicial determination of ineligibility is a removal from Office.