I believe I have it figured out, at least as applies to my situation.
I have an ad blocker running on my router (Tomato firmware), which blocks many blacklisted/sketchy hosts. Among those is doubleclick.net.
Well, it turns out that as PBS recently added adverts to their Roku app, starting with the most recent 2.7 b3, one of the ad servers it's querying is doubleclick.net
Guess what happens when the router gets that request from the Roku? It blocks it, which in turn causes the PBS app to crash to the home screen.
This behavior is 100% reproducible if the Roku is blocked from access to doubleclick.net
Bottom line: No, I am not going to whitelist doubleclick.net just to get around this crap. This is fundamentally an issue with the app which needs to be addressed. The app needs to respect my choice to protect my entire network from sketchy and crappy sites like doubleclick.net, and it needs to learn how to deal with the fact that it can't reach a particular server, instead of spazzing out and crashing.