I supposed not impossible, but genuinely remarkable. Just applying simple logic, the only way the known facts of the case work is if the killer was not afraid of being caught. It's the only logical answer.
If the killer was brazenly in the house while the Ramseys were out, and busied himself writing that preposterous ransom note, it says he was not afraid of getting caught.
If the killer wrote the note and did everything else while the family was asleep upstairs, it says he was not afraid of getting caught.
It is an extraordinarily rare criminal, indeed, who is not afraid of getting caught. A stranger so psychotic as to do all of that without the slightest care for his own self-preservation is remarkable, indeed. Is it impossible? No. But it would be astonishing if that's the way it went down.
I agree that this case is very unusual and there are lots of unanswered questions. Like:
How is it that matching DNA not belonging to a family member came to be under her finger nails and on her underwear on the same night that she just happened to be murdered?