I think there are a number of things that come into play to answer that question.
The first, most pressing thing related to your network connectivity: There's absolutely nothing about how Channels connects from the client to the server that is any more elaborate than anything else. We make simple HTTP connections over a local network. What we're seeing in your logs is that the HTTP connection to your DVR just doesn't respond. There's nothing that we would be doing that would cause that to happen or not. It's just using very basic OS APIs. So... maybe your network just happened to be flaky this week? Solar flares? You had a new neighbor move in with high powered wifi on the same channels? Someone got a new microwave? Hard to say.
More generally, why would people say that Channels can benefit from better hardware than some other solutions? A lot of that is because you can do things in Channels like TVE that you don't really have with other solutions.
The Channels DVR can scale way down to very little hardware requirements up to fairly high requirements based on usage and need. Just recording 1-2 shows at a time and watching 1-2 things at home at once from an HDHR and never leaving the house? You could be great with a Raspberry Pi.
Want to remotely stream a bunch of HDHR cable streams live while also recording some TVE? It could require more to do that well.
But, your expectations that you should be able to throw this on a SHIELD and have it work are definitely in line with what we generally see as an experience. These timeouts are unusual and shouldn't happen on a SHIELD.
