Converting entity to JSON
View builders only display HTML/render arrays.

Look into the serialization/rest/hal modules if you want to extract the data of your entity, as you guessed already. 

Note that REST in core only supports content entities at the moment, but I assume you have one of those.

Also, I'd recommend hal+json serialization, provided by the hal.module instead of the raw json seralization, as that can't deal with things like references properly.