If you recall Jojo's Bizarre Adventure, only people with stands can see other stands. Snuffy is Big Bird's stand. The other people don't have brain damage(as far as I know), they just didn't have a stand yet.
DID U KNOW THAT SNUFFY IS ACTUALLY A FIGMENT OF BIG BIRDS IMAGINATION BUT LATER ON IN THE SHOW EVERY ONE STARTS BEING ABLE TO SEE HIM; THAT MEANS EVERY ONE ON SESAME STREET IS DELUSIONAL OR HAS BRAIN DAMAGE
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