Idol Manager actually used to be "turn based" in its early stages of development, years before the Kickstarter. Here's a gameplay video from 2016 showing how the game worked back then.
Interesting... I wonder how much of the internal code has changed since then. Personally, as a coder, I'd love to mess with the time buttons to see if I could make a turn-based mod for the game. However, I don't know if there's any way to re-compile the dll without the original source code, so I might be stuck with wishful thinking and/or making a similar system and playing with that.