You can make it by setting up several schedules.

Sounds strange. Able to control via HomeAssistant but failed to control with phone? Did you open VPN service on your phone? Make sure your phone is connected to same 2.4G network.

When the devices aren't in the device list in Google Home App, can you remotely control your devices on Kasa app?

Emmmmm， I also notice this problem. Seems that latest Kasa doesn't support unsecured network for some safety issues. If you use the old kasa version , maybe you can configure your plug to open wifi.

Surly you can, but you should enable remote control function first.

Try to foget your router's wifi and then reconnect to plug's wifi, and make sure you are close to the plug. Also, did you try another phone or tablet? The plug's wifi is same with softAP.