Really there were two Letterman bands led by Paul Shaffer: The World's Most Dangerous Band on
Late Night with David Letterman 1982-1993, and the CBS Orchestra on
The Late Show. They had some of the same personnel but two different vibes. I vote for The World's Most Dangerous Band.
Because Johnny Carson owned the show, he laid down rules so LNwDL would not be seen as something that could easily replace
The Tonight Show. One of the rules was that bands could not play on their own. They had to have a member or two sit in with Paul and the band (more members were allowed as the show went on, but still with the house band). If the show really wanted a band to perform on their own, they had to get special permission from Carson Productions in California, and that was rare. Two bands I remember being allowed to play on their own in the early days were R.E.M. and X.
This was not easy. In the old days there were full time professional bands/orchestras and musicians unions in big cities and singers could bring their arrangements with them and everyone spoke the same language and they could put shows together very quickly. This was often how Doc Severinsen and the NBC Orchestra worked. In the rock and roll era, that professional establishment kind of faded away. Rock and pop acts in the '80s were used to performing with the same touring band. So when they played the Letterman show, they had a couple of hours to get everything together with a band they had never met before. The "Late Night" four-piece was recruited from some of NYC's top session players, and was so professional, talented and musically versatile, they could go from country to folk to funk to metal in the course of a week, and hit the spot on each one.
Also because 6A was so small, relatively, the band were probably less than ten steps from the host desk. Because they were so close, they were part of the show even when they were not playing. SNL veteran Shaffer obviously took to the comedic sidekick role from the start, but the other members (Will Lee, Steve Jordan, Hiram Bullock, Anton Fig replacing Steve, Sid McGinnis replacing Hiram, and part-time member David Sanborn) were too close to ignore. Dave talked to them casually during the show and if they had something funny to say, they said it. I've never seen another band that had the same easygoing work-family feel.
Paul is a rock-and-roll encyclopedia, and if he thought of some music that was appropriate on the spur of the moment, they could be on it within seconds. Because they became so well-known and respected for being able to play with anyone, the WMDB became the house band for the Rock and Roll Hall of Fame induction, and Paul led the closing jams on stages filled with rock and pop legends.
One memorable example of the band's greatness is this 1986 performance by Steve Winwood. Trying to replicate the "Higher Love" studio record was a challenge, and Paul tries to warn Dave at the beginning. The time changes of the instrumental breaks are tricky and change on a dime. Anton does a monster job timing the performance, backed up by highly respected studio percussionist Carol Steele. You can see Paul and Will are
glued to their charts. And yes, Paul plays a clinker. I also have to mention the wonderful Dolette McDonald on backing vocal.
Last-minute adaptability: Guest Gary Busey plays an impromptu number with the band after the musical guest (the Charlie Watts Quintet) cancels. Relevant part starts at 1:36:20.