What you're seeing with the light is normal - it goes off when the screensaver activates. You may just need to pair the remote again. My preferred method for pairing a remote is to press and hold the pairing button in the battery compartment until the light starts flashing, set the remote down next to the Roku device, and only then connect the power to the Roku device. When it finishes booting hopefully it will pair with the remote.
The Roku 3 and presumably the remote you're using are getting pretty old at this point, so it's possible for one or the other to be bad. When the Roku 3 displays the Home Screen do you see the clock in the upper right hand corner or "Not connected"? If the clock is displayed then it's connected to the Internet, so that would point to the remote. If "Not connected" is displayed then that would point to the Roku 3. If the clock is displayed then you can try the Roku app on your phone/tablet and use it as a remote or get a cheap universal remote that supports Roku devices.
For the Roku app: Roku Mobile App - Free for iOS® or Android™ | Roku
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