You're overthinking this. One reason is money. They get two main characters for the price of one. Second, in the beginning of Sesame Street, there weren't alot of Muppeteers, so as minor muppetz pointed out, they all doubled and tripled and quadrupled on characters. That's the way voices are cast in animation as well. Screen Actors Guild rules allow one performer to do two voices for the same fee. Plus, if you've read any of Carroll's writings, he was afraid he was going to wash out as a Muppeteer in the beginning. I doubt Jim would want to be training two people from scratch for this important show.
Next, it's not that complicated technically. I'd guess they have someone else performing Oscar, while Carroll just changes his voice in the Bird suit. They're not ad libbing. The Oscar puppeteer would know what he was going to say to move the mouth.
And finally, scheduling. Jim and his core group basically worked one month a year on Sesame Street, taping all their bits so they were free to do other things the rest of the time. But Carroll had to be available for all the street segments. Why not have him do the two main characters that interacted with the humans? It makes perfect sense.