The power to challenge the "behavior" of the President belongs to Congress not the courts.
Exactly the argument I would make. Its not that the president is immune. Its that the place to charge him is the House and Senate as set forth in the US Constitution....and don't tell me that doesn't apply to former presidents because the Democrats ALREADY DID SO with their dog and pony show second impeachment of then private citizen Trump for the Jan 6 mostly peaceful protest.
If the SCOTUS does not rule that the House and Senate is the place to impeach and try a president or former president, then it will be open season for every partisan DA in a partisan district to bring criminal charges against a former president of the opposite political party. For example, a DA in a deeply red district could bring criminal charges against former president Obama for killing American citizens away from a conflict zone with no due process of law.......Does the SCOTUS want to usher in a series of tit for tat prosecutions by highly partisan political hacks of every former president? If they do not hold that the House and Senate is the place to do so then that is exactly wha they are going to get.