Quick update here on my end... after installing my most recent TV update a few days ago, it appears that the apps are launching normally again. It's no longer dependent on what TV input is opened, as I mentioned earlier in the thread, and so forth.
I'm still not sure what caused this originally or if it's something that might pop up again in the future. But perhaps the last TV update or two had something that didn't mesh well with my router or the specific TV model? In any event, it appears to be fixed now, so hopefully it stays that way in the long run!
If it reverts back to the old situation, I'll keep you posted. But if nothing else is working, just keep in mind that future TV updates could end up solving the problem.