FWIW I was able to confirm what you(OP) are seeing. I disconnected my Modem from my Router. My network was still all connected just no access to the internet(WAN). I checked the Tablo access via my Roku Ultra wired via ethernet and I was still able to access the Tablo menu. However on a wireless connected Roku Streaming Stick when attempting to access the Tablo there it would not connect and got a message “no internet access.”
I’m going to look at a few other things mentioned by zippy to see if I can narrow this down.
My Tablo Quad is still on the 2.2.28 firmware.
Update: Tinkered with it some more. I was able to verify that my Tablo was still connected to both my main Router and my AP(Access Point) even when the internet is disconnected. Again I am able to access my Tablo even with internet down but only with my ethernet connected Roku Ultra. I tried 2 separate Streaming Stick +s(wireless connections) and neither one would connect.
At this point I am not sure if it’s the Roku devices that is causing the issue or the Tablo software via those Roku’s.
Oh and just an FYI but there is a long thread on a similar topic. Just now reading it all.