Well, I watched every episode on YouTube, and it seems that the band slowly became a major part of the show. Digit is the most major character in the group, due to his prominence in the controll room and since he appeared in every episode (that had MuppeTelevision). Out of the band Clifford seemed to be the second most important character in the band, appearing frequently on his own.
Beard, Flash, and the drummer all didn't appear until the thrid episode, and even then, they wer ejust in the background. The band performed back-up in the closing number, but after that, they weren't seen as aband until the Smokey Robinson episode, when the band provides back-up in Smokey Robinson's first number, appear backstage before the number, and also apear in the closing number (though the drummer was absent from that).
Then the band got their big break in the enxt episode, having a major presence in the controll room, and having their one big hit, The Music Just Keeps Us Rolling Along. This is also the only epsiode to really develop Flash as a character, and the only one where the drummer talks.
Then, in the k.d. lang episode, Flash, Beard, and Clifford provide back-up as a country band in k.d. lang's first number, and are joined by Rowlf. I guess Digit wasn't there because of Rowlf, and who knows why the drummer wasn't there. Then in the "food" episode, Flash and the drummer are replaced with Zoot and Animal. I'm not sure whether this was intended to be a permaneant change to the band or not, but the band is heavily featured, appearing in most of the controll room scenes, taking "requests" from Kermit, and performing in teh closing number (so I guess then that they did have more than just one main song).
And in this episode it seems like they were trying tod evelop Beard as a main character, and judging by that episode it looks like he might have been the leader. Beard was also a little bit major in The Secrets of the Muppets. He wasn't that major, but he was the only member of the band to appear when the puppeteers are shown with their characters, and besides Digit, he's the only one who appears in the special (outside of the behind-the-scenes footage from The Music Just Keeps Us Rolling Along).
For awhile, I had thought that The Muppets at Walt Disney World should have had the whole Solid Foam band there, perhaps forming rivalry with The Electric Mayhem. But as they weren't very major until the end, I guess it makes a bit of sense that they weren't developed after the show ended.