I reviewed the Rams' initial police interviews and the 911 call. JR first read the note when he joined PR downstairs after checking on Burke. PR dialed 911 at 5:52 AM and told the operator they had a ransom note while JR was still reading it. The call lasted 1:13 min. PR began calling the Whites and the Fernies immediately afterwards. The first police officer arrived at 5:59. So the short answer to your question is that John had no way to get rid of the note even if he wanted to.
If you believe the Rams. Alternatively, and more likely in my view, the writing and reading took place much earlier (more on this later). However that process went - collaboratively, PR only, several drafts, etc. - and whatever JR thought of the final version, he knew they only had so much time, and that an iffy RN with clues to an intruder and motive was still a safer bet than a 10-word note or none at all. He might even have thought the rambling mix of elements sounded suitably deranged and could work to their advantage. He figured out (maybe was advised?) that they didn’t need to stage/tell a story so good that it virtually proved their innocence, only one good enough to make the police unsure of their guilt.
JR is intelligent, but it's possible to overestimate his judgment at the time. I mean, he's the one who gave police the cringey story about breaking the basement window months earlier and stripping down to his socks and skivvies to climb in…..and also said he was wearing the same outfit when he read the RN. Wait, what?
Maybe JR was in his socks and undies while reading the note, and that experience colored his thoughts when he made up the basement window story? JR says he was in his bathroom shaving when he heard PR scream and ran down the back stairs to the 2nd floor. Later, police found his bathrobe on the floor of his study, perhaps cast off on his way to the shower at 5:30. But, he would have to have gone back through the study to reach the stairs. It was December. Why not grab the robe on the way, especially if he didn’t know what awaited him or how long he’d be downstairs? The dots don’t quite connect.
I have a theory that the reading-the-note story is as fake as the basement window story, and that the reason the socks and underwear image kept intruding on JR’s thoughts was that it was trauma-related; that he actually was dressed that way at some point during the night, perhaps when he was molesting JBR, or when he didn’t want to get fluids or fibers on his clothes as he cleaned up evidence and staged the discovery scene.
I' reply to your later post as soon as I can : )
