I feel like I'm missing something obvious, but is there a Linux client similar to the one for the Shield? I've got a raspberry pi I'd like to connect to the Channels server on my PC.
No, there is no Linux client.
There is a Kodi PVR client of sorts, but as far as I'm aware they have not made Linux versions of the addon.
How about using the web browser?
That's probably the best alternative. I really like the client on the Shield. Very user friendly
When I'm viewing on one of my computers, I either use the web UI, or I bypass Channels completely and just open a stream directly through mpv.