No, there should be no interaction between the Smart TV and the Roku. I have multiple devices connected to the same TV, and Prime works fine on all of them. It sounds to me like the Prime channel is corrupted a bit. The first thing to try is to remove the channel, reboot the Roku (critical step, do not skip), then add the channel again and log in. I've never had to log into Prime once it was installed on any Roku, even after several years of use.
Dan
Roku Community Streaming Expert
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