Does not seem to preclude it originating as a series of verbal tales.
According to the article, doubts about single authorship began to be raised in the 19th century. This was a time when a lot of people thought that they were living at the pinnacle of history, and that cultures of the distant past must have been strictly inferior to the modern one. Troy could not have possibly existed; no single person in such a barbaric age could possibly have produced great poems like the Homeric epics -- or Beowulf.
Well, we found Troy. We also found evidence of great devastation at Troy right around the time when the Homeric war supposedly took place. It seems that people of the distant past did possess the ability to tell a great story after all, moving freely between history and mythology, filled with allegory and philosophical depth. Just like Tolkien did, but hundreds or even thousands of years earlier.
It's hard to imagine that that idea could have fallen out of favor, given the incredibly sparse written record of the Germanic tribes. They didn't produce enough literature to have a non-oral tradition.
Which suggests that some of the perceived homogeneity may stem from the scribal style, rather than the oral style.