Following up on your scenario .... He would have had hours while the Ramseys were away that night to roam around the house and more time after they got home and he waited for them to go to sleep, to write the note.
It's "noise" which awakens our instincts. Besides, during the holidays, things get moved all over the place.....and they have a housekeeper.
where did he stay the night before the crime, and where did he go directly after the crime?