In conversations with colleagues, fellow entrepreneurs and even musicians over the past decade, Daniel Ek would often abruptly shift the subject to something that really bugged him: health care. “I was like adamant to fix it,” Mr. Ek, the Spotify chief, told DealBook. He saw the industry as a bloated and inefficient colossus in need of disrupting.

The problem: Mr. Ek had neither a plan, nor the time or money to do much about it. He was busy taking on Apple, YouTube and Amazon Music in the streaming wars. In his spare time, Mr. Ek pored over medical journals. And he routinely measured his vital statistics with a Fitbit, an Apple Watch or Wii Fit tracker — the more data, the better to see how his body held up against the rigors of…