I'd say: not as long as anyone remembers what happened to Nixon. That seems to be the one use of memory that Trump employed: when meeting with the Russians from the start of his administration, he refused to even have the standard State Department witness in the meeting, and forbade the interpreters from writing up their notes.
OTOH, don't think about employing LBJ as a precedent for shielding his privacy: he seemed to have absolutely no sense of privacy. When Doris Goodwin was an intern she remembered he'd invite her into bed so he could answer her questions; she gave no other details! He used to continue his business meetings in the presidential "head" when he was obliged to dump on more than just his subordinates. I believe VP Humphrey was one such victim. He used to boast about the details of his "member" in settings that no other man would.
After the Biden White House gets done cooperating with the House committee investigating the Trump insurrection, who knows how that would thin the future herds seeking the presidency?