There a re a lot of reasons...the overexposure bit is the one I thought of first. Even Ed Christie said that about the character in an interview (I can provide a link to anyone interested). DrTooth is right about letting Elmo emote and become more than just a giggle. Give me the old "Happy Caps" episode any day...I wish Sesame Workshop would put that online!
For those of us who have "Street Gang", Jon Stone was frustrated (and rightly so) about the show being "dumbed down" to compete with that horrible purple...He-Who-Must-Not-Be-Named. I wish that I could read Stone's memoirs...
As for the purple tumor, Caroll Spinney mentioned in an interview that the other show really hurt Sesame Street because they were better at selling toys than Sesame.
JustinHuskie, you might find this thread interesting. It's old, but it might answer your question to some extent.
http://www.muppetcentral.com/forum/threads/my-theory-on-elmo.18087/#post-348650
You might also want to read the thread about "The Street We Live On", a special which aired some time ago:
http://www.muppetcentral.com/forum/threads/your-thoughts-the-street-we-live-on.12060/
Just something to add, the popularity of Elmo also hurt those who call themselves fans of Jim Henson and want to be taken seriously. The era between the end of "The Muppet Show" and let's say the Muppet video of "Bohemian Rapshody" was a difficult time for Muppet fans. During that time, the regular Joe would see two things regularly on TV associated with Jim Henson: Muppet Babies and Elmo.
So therefore, the casual fan had this set of simple equations:
Jim Henson="The Sesame Street Guy"
Sesame Street=Elmo
Muppets=Muppet Babies
Elmo+Muppet Babies=kids' entertainment
Therefore:
Jim Henson=a preschool entertainer and nothing else
Now, the Muppets=Muppet Babies reasoning isn't as true as it used to be. This is in part to the viral videos, the latest film and even Muppet Treasure Island helped to sway the adult audience back to the Muppets. Also Muppet Babies hasn't been on TV in a while.
Sesame Street has never taken a break. It's always been on since 1969. However, since Elmo's rise to popularity in 1993-ish and onward, he's guaranteed to be in every single episode. If Elmo is given the chance to be more of a three-dimensional child, such as Big Bird, then the deeper, more eloquent lessons about life can be taught on Sesame. The Good Birds Club episode was a
brilliant lesson in self-acceptance, pride and not giving in to bullying. Kids will benefit way more than this than 15 minutes about "hands" or "hats".
Kids are smarter than what we give them credit for. Leaf through the catalog of Joe Raposo and Jeff Moss songs and you will find an array of beautiful music dealing with all kinds of topics in a wide variety of music styles. This is just one aspect of Sesame Street's brilliance. (While you're at it, please read the poetry books of Jeff Moss..."The Other Side of the Door" and "The Butterfly Jar"). These pieces educate, entertain and encourage kids to think, wonder and discover.
Compare this to one word sung over and over again to the tune of "Jingle Bells". I agree with what Frank Oz said about Sesame Street a while back...because let's face it, he's right.
Sesame Street did wonders when it came to teaching subjects of basic humanism without insulting or talking down to the audience. Sesame Street was designed from the beginning to introduce kids to the world, to teach them about things they didn't know.
Even little kids already know what feet, hands or any of the other Elmo's World topics are. In Sesame's "magazine format", when a song about one of these topics would pop up, it was OK. It was once in a while, not a 15 minute block of time every single day.
You'll find that there's a lot of people here who have a lot to say about "the red menace".