Does Breath of the Wild for Wii U require a TV?
Solution 1:
Having a Wii U, TV, and BotW, I can say that one can play BotW on the Wii U without ever needing to connect to a TV.
During the first cutscene, it will assume it is connected to a TV, but there will be a textbox on the gamepad screen that says, "touch here to play on the gamepad," or something like that, and it will pull the video down onto the gamepad for your playing pleasure (something that allows me many a long night of gaming without waking the entire household).
However, keep in mind that if you touch the gamepad screen, it will revert the video back to the "TV" and you will need to tap inside the box again to bring it back.