Let's not forget too that Bert has had a handful of solo songs over the years as well; "Doin' the Pigeon" and "An Oatmeal Box" immediately spring into mind.
Then there's also songs he did with other characters who weren't Ernie, such as "Take a Rest" with Frank's other two characters (makes me wonder who puppeteered who in that one), and "Pigeons, and Cookies, and Trash" with Cookie Monster and Oscar.
Either way, when you consider Ernie and Bert were conceived specifically to be a comedy duo, it only makes sense that neither of them really work as well as a solo character . . . as with other comedic duos, you can't really have Abbott without Costello, Laurel without Hardy, Kenan without Kel. Ernie works better as a solo character only slightly more than Bert because Ernie better conveys childlike wonderment and whimsy that kids watching can relate to and identify with . . . Bert, not so much. That, and one reason too why Ernie works well with other characters aside from Bert is, when you think about it, Ernie would be paired up with another Frank character like Cookie Monster or Grover, so it's still Jim and Frank playing off of each other as they do so well, regardless of what characters they're playing: Ernie and Bert, Ernie and Cookie Monster, Kermit and Grover, Kermit and Fozzie, Kermit and Piggy, etc.