Same set up. Same problem. Wish I had found this thread after spending hours working on this myself and then with mlb.tv phone support. Tried everything including full Apple TV reset. Many hours of failed troubleshooting. Ugh. Ultimately I just canceled my subscription and requested a refund. MLB didn't seem very interested in getting it working for me. Oh well.
Now that I see the solution in this thread, it makes complete sense. The audio routing to my HomePod makes complete sense as the culprit. MLB is so nuts about people stealing their content (no Airplay support); you can't even plug an iPad Pro directly to a TV with an HDMI cable to watch. They block that in the MLB app too. That's a bit over the top if you ask me.
Too much hassle. Bye bye MLB.