@BurrittsRapids 

I would like to double confirm with you, are you using the Schedule feature of the smart plug? Or did you create an automation rule to control the smart plug?

Automation is a cloud-based feature and it cannot work without internet connection. 

How to setup schedule, away mode and timer of TP-Link Smart Plug/Switch in Kasa APP

  @BurrittsRapids 

Instead of controlling the router, try switching to a different appliance to test this feature first, such as a lamp. 

Disable the internet access for this Kasa plug through the router, then create several schedules with short intervals to observe if it takes effect. 

  @Ttandat 

Is your device's time zone accurate?

You can try a simple test by setting two schedules with short intervals that will take effect in the future. After the first schedule takes effect, turn off router and observe if the second schedule still activates.