I don't exactly understand why people are against the Sleep Advancement system. I imagine it would work somewhat like the Poke-Palago from Sun and Moon. If I were to implement it myself, it'd probably work like this:
In the castle/town builder you can make certain facilities used to harvest various resources (I'd assume there's also more to do in the town but I haven't the faintest of what they are). Something along the lines of Mines for refining stones, Colosseum for arena medals, Hen house for feathers(?) and so on and so forth. Once the area is built, you can assign a team of up to four units to go in and harvest materials in the background. You can't use units while they're harvesting, but the incentive to use your stronger units is that they have a higher harvest rate. Let's say... 4 Star units can get 1.5X as much materials that a 3 star can collect, and 5 stars can get 2x as much as a 4 star can.
You can choose how long you leave units harvesting, anywhere between 8-24 hours. The longer you leave them, the more materials they can collect. If you leave a team of 3 stars in the Colosseum for 8 hours, they can collect 16 medals in total, four for each of them. Leave them there for 12 hours, and they collect 30. Leave em' in there for 24 hours and you get 70 medals. If you left full team of five stars in there that would be a whopping 210 medals for absolutely no work at all. Looking at it now, there's no way that we'd get that much for free, but I hope you get the idea.
I'm pretty sure that you don't have to leave the game closed for a long time to get any effect out of it. But I'm assuming everyone here sleeps at least a bit, and if you plopped some heroes into the mines while you sleep, it's never unwelcome to wake up to a few materials imo.