As a teenager Debbie Greenberg was spending far too much time at the Cavern Club in her hometown of Liverpool, England. It was already the most famous music club in the world, where she had been dazzled by the Beatles’ debut performance and had witnessed their rise to stardom for two years before watching the local heroes leave home.

Then in 1966, after the previous owner declared bankruptcy, her father asked her out of the blue if she thought it would be a good idea to take over the club. It was an offer she couldn’t refuse. She’d been a Cavern fanatic since it...