When you have Not Connected on screen it's not referring to your remote, it's the internet that's not connected. Reason the tv controls still work it's being controlled by IR and has nothing to do with Roku itself.
If I had to guess what's wrong? I would first have to assume you have 3810 stick+, it's usb cable is allowing power through, but the long-range wireless receiver, the bump in the middle of the cable is bad. That's where the Internet and Remote communicate with the Roku.
Edmund
Roku Community Streaming Expert
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