From within the Roku TV, maybe try enabling Fast Start. (Settings..System..Power..Fast TV Start Enabled)
From router, you can try extending the lease time to the assigned MAC Address of the Roku TV. Can also try to assign it a static LAN IP to bypass the lease time.
Then there is the physical replacement of the ethernet cable itself (or at least the end connectors rj45).
I think the problem is with the NIC, not the other possiilities above. It is the right one, no problem there, but, as you've seen, I think it is being powered down by the USB port when you turn off the TV, thus losing the network connection. (Everything is operating as designed.) I would try enabling Fast Start as the first step. This should place TV more in a Sleep Mode, than Off mode.
Other than that, I know of other other way within Roku settings to keep the USB port powered.
Roku Community Streaming Expert
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