Wow indeed! That is some mighty router. It's the first one I've ever seen with more than 4 Ethernet ports. Nice! Based on the documentation I found, if you are using its default settings it should be using the address range 192.168.1.x, so you're good there. Roku shouldn't be thinking it's a public network.
Linksys doesn't seem to have a comprehensive manual available, so I can't say for sure what options are available in the interface. But you want to ensure that wireless connected devices are allowed access to your local network. Some routers have a setting to only let wireless devices reach the Internet, but nothing within the network. I can't say how they might have such a setting labeled. But if it has a guest wireless network enabled, that too would block access to your internal network, so you can check that as well.
Nvidia Shield, Roku Stick (3600), Ultra (4640), Premiere (3920), Insignia 720p Roku TV, Sharp 4K Roku TV, Windows 10 Pro x64 running Serviio and Plex on a wired Gigabit network.