In some respects, the members of the Embarrassment defied the odds. The band blossomed in the dry cultural soil of Wichita in the late 1970s and early 1980s, helped birth a music scene, toured the country and won the respect of some of the most influential artists of the day. But the big break never came. Ultimately, the major record labels and influential radio stations didn’t see much potential in a Midwestern art-rock act with songs about British mathematicians and patio furniture. “We were from Kansas,” said frontman John Nichols, “and we looked like nerds.” In 1983, after five years as a band, the Embarrassment called it quits. But the fervent fandom that had surrounded the group never really died out. Over the ensuing decades, the Embarrassment became a kind of secret handshake among record collectors, music critics, local radio DJs, cool older sisters and other zealous pursuers of the talented and obscure. (Robert Christau, the so-called “dean of American rock critics,” call...