I have an enhanced remote that came with my Roku 4.
My Roku 4 is connected to my router with an ethernet cable.
I turned off the wireless signal from my router.
My questions follow:
Can the remote connect and control my Roku 4 since the wireless radio on my router is turned off?
Is the remote designed to wirelessly connect to my Roku 4 directly or to wirelessly connect to the Roku 4 through the router?
Roku point-anywhere / enhanced remotes will communicate with your device on their own direct wireless connection. In your case, the remote should be able to pair to your Roku player and work normally.