Whole home humidifier suspected of causing water softener problems
I think you've hit it on the nail. The humidifier takes a slow trickle of water, too slow to turn that impeller. This leaves the softener thinking no water is being used, when actually lots of water (over time) is being used. Eventually the media becomes saturated long before the softener decides to regenerate it.

Ion-exchange water softeners absorb calcium and magnesium and replace it with sodium. So, it seems likely that your humidifier will get gunked up no matter which side of the softener you connect it.

Edit: Is there a way to externally and electrically trigger a softener recycle? If so, then perhaps you can have a timer driven by the humidifier that, once every other day of humidifier runtime, triggers a recycle. Make sure that the softener does NOT perpetually recycle if the timer stops at just the wrong moment.