As someone who once had a toddler, he's not beloved, he's hated by most parents. Because he's fucking awful. Stop talking to the fish and the baby, Elmo! They can't fucking understand you! Also, you don't seem to notice that Mr. Noodle keeps changing and that lack of basic attention to detail is worrying.
On top of that, his parents basically neglect him, which is why you never see him.
Fuck Elmo. Give me Cookie Monster, Ernie and Bert and Oscar the Grouch.