You do that until the answer to the original question no longer matters.

That rarely happens. Not the "raising every possible argument and appealing at every turn" part - that happens all the time. But it's rare that you can do that until the answer no longer matters.

Trump's immunity claims ended up delaying his trial by only about six months through procedural matters. Because he won, it will cause a lot more delay. It's only because of the extremely rare combination of circumstances that the prosecution brought the charges two and a half years after the election and that defendant in this case is likely to win the Presidency that a six month delay is going to matter. If there had been a case at the end of the terms of Clinton, Obama, or Bush (either one) the answer to the original question would not be mooted by the review.

Most importantly, the status quo would stay - Trump would remain in Gitmo.

No, he wouldn't. The district court would immediately order his release - and unless that court or the appellate court stayed that order pending appeal, he would be released immediately. Just because you appeal something doesn't mean the status quo remains in place.