Well, you insist upon imagining things that in the real world cannot happen. But, let's make this easy and assume that Vladimir Putin did somehow become president as a result of a violation of the Constitution. The only constitutional way to remove a president is through the impeachment process.
Returning to the real world, though, one of the issues in the campaign of 2008 concerned the claims expressed by some that Obama was not a natural born citizen. By selecting Obama, the voters and their electors implicitly resolved those expressed claims in Obama's favor. Had you been an elector, you might have decided the issue differently. Okay.
Seriously, what do you think should be done at this point?
SCOTUS should hear the cases about Obama’s eligibility, subpoena all the records, and if it is found that he is not eligible he should be judicially disabled from “acting as President” as per the 20th Amendment. Joe Biden should immediately become “acting President” and be impeached for treason. Nancy Pelosi and all the other Congress-critters who knew Obama was ineligible should be arrested immediately for treason, after which impeachment hearings should be held for Joe Biden - with the one or two Senators who aren’t in jail.
No they didn't. They acted out of ignorance. (Good job media!) Do not mistake blundering about blindly for affirmative selection.