When Ben Drury was shopping for his newborn at a department store in London, he saw something that made him grimace: A stroller on display had a holder for an iPad to keep an infant occupied.

“I remember having this visceral reaction against that, saying, ‘No way, this is really bad,’” he said. In that moment, Mr. Drury wondered whether there was a technology product he could create that was enriching for children but not addictive.

That was 12 years ago. Around the same time, Mr. Drury, a digital music entrepreneur, also realized that with cassette tapes and compact discs now long dead, the simplest way for parents to play audio for their children was to open an app on a phone or tablet with a touch-screen. A more developmentally appropriate gadget would have physical controls to help little ones learn to use their hands.

So Mr. Drury, now 50, and his business partner, Filip Denker, 49, another newly minted father at the time, got to work creating a product that they felt parents desperately needed: a music player for children.

The device, which would eventually be named the Yoto Player, is a rubber-encased box that emits audio from a speaker when a child inserts a card into a slot, similar to a retro Nintendo that reads game cartridges.

With millions of devices sold, the Yoto Player has since gained a cult following. Hundreds of its fans have even crafted Yoto accessories sold on Etsy, including Yoto card organizers and cup holders for carrying the player in a car seat. A competitor called the Toniebox, another screenless audio player for children, is also gaining traction.