When, in our 200 year history, was criminal immunity required for the President to make the correct choice for the country?

perhaps a more recent example - President Obama asserted executive privilege to prevent the disclosure of various government records to Congressional Republicans during the "Fast and the Furious" investigation. That assertion of executive privilege was later overturned by a federal judge, who ruled that Obama had an obligation to provide Congress the documents. Doesn't take much of a stretch to argue that President Obama was engaged in a conspiracy to obstruct a federal proceeding by wrongfully withholding those documents - for the purpose of improving his personal political power to boot.

This is the normal give-and-take between the executive branch and the legislative branch. The judicial branch intervenes when needed as a tie-breaking vote. Criminal immunity is not needed. (The "Fast and Furious" DEA operation was a disaster. The House held Attorney General Eric Holder in contempt of Congress, but the DOJ did not pursue prosecution. Congress did a long investigation, but did not change any laws because of "Fast and Furious".)

We've never had to parse that out before, because prosecutors have never gone after a former President for actions they took while President. But I don't think it would be too difficult to find a "conspiracy to defraud" charge against every modern President, because every modern President tries to hinder Congress from doing something.

Nixon, Reagan, Clinton, and Biden were gone after by prosecutors. Nixon was the only one who asked for criminal immunity.

Conspiracy against the United States under 18 U.S.C. § 371 requires criminal intent. The President would need to prevent Congress's normal functioning by means that are dishonest. I can't think of any examples of this where the country benefited. Bush may have lied to Congress to start the Iraq War. Trump tried to stop Congress from certifying the election. Neither of these was good for the country.

Conspiracy against the United States
"To conspire to defraud the United States means primarily to cheat the government out of property or money, but it also means to interfere with or obstruct one of its lawful governmental functions by deceit, craft or trickery, or at least by means that are dishonest."
en.wikipedia.org - Conspiracy against the United States

Trump's brief to the USSC lists some historical examples. Most are easy to dismiss as being bad for the country. (These should not be encouraged through criminal immunity.) The drone strikes are covered through separation of powers (the president is the commander in chief). I haven't seen any example when criminal immunity was required for the President to make the correct choice for the country.

"American history contains no shortage of examples of Presidents committing allegedly “criminal” official acts—at least in the eyes of their political opponents."
John Quincy Adams appointment of Henry Clay as Secretary of State.
Andrew Jackson infamous “Trail of Tears.”
President Roosevelt internment of Japanese Americans.
President Clinton "Wag the Dog" military strikes.
President Clinton pardon of Marc Rich.
President George W. Bush lying to Congress to start the Iraq War.
President Obama drone strikes.
President Biden’s management of the southern border.
President Biden’s dealings with Iran.
President Biden’s funding of international groups.
supremecourt.gov: PDF document