Smooth sailing here with MLB.TV. Thanks to everyone that helped, especially hoo.
I hadn't looked at this thread in a while and so hadn't seen this comment. Glad to hear my advice helped. MLB.tv can be finicky at times. More than once this season I've had problems with it and the solution I found online that works, no joke, has been to delete the app from my Apple TV, pull the power cord, wait 60 seconds, power back up, and reinstall the app. Weird stuff.
But I think it's not just MLB.tv, either. We had trouble with the NBC Sports app last week when the Caps were on NBC Sports Washington Plus. YouTube TV doesn't carry that channel, but YouTube TV subscribers can watch the local feed via the NBC Sports app. But for whatever reason, it failed to load on both our Apple TVs and our Fire TV devices, yet it worked fine on my iPad. Watching a game on a 9.7-inch iPad when there is a 40-inch TV across the room burns me up, though. (We have a 55-incher downstairs, but the seating configuration there is less conducive to watching on an iPad.) Eventually I got the game to load on the TV for the second of the two games that were on the alternate channel, though I don't remember how. I know for the future there is a way to connect an iPad to the TV, but I'll wait until closer to next season in case the YouTube TV channel lineup changes in the interim.
What it all underscores to me, though, is that there are always hiccups with streaming your TV and that annoyances will crop up seemingly at random. One thing the Caps experience underscored to me is to keep your mobile phone handy—the NBC Sports app required either entering a code via a web browser or scanning a QR code, either of which is to authenticate your subscription with your participating TV provider. Obviously the QR code is the easier option.