It’s been some years since I read in any depth about the case. If memory serves Charles’s possible culpability was an intriguing alternative theory that helped to explain some of the discrepancies in the case, as well as his own behavior — previous shenanigans with the child, crying “they took our child”, finding the note himself, ordering everyone out of the nursery between 8 -10.
I don’t recall if Hauptmann was thought to be Lindbergh’s accomplice, or an opportunist who collected the ransom. Certainly Hauptmann was involved in some way.
I think the argument against Lindbergh covering up the baby’s accidental death is the degree to which he became involved between March & May (when the baby’s body was found) in running down one lead or another — knowing as he must, they were all false. If memory serves, he went far beyond a few face-saving efforts, to a degree that suggested he really believed his son was alive & that he could recover him.