My only option to play Zelda is to buy a Japanese Switch and a Japanese copy of Zelda, but I'm concerned not every game will change to English if the system language is set to English. I know in the stream it seemed like every game that Jeff played from Japan was in English, but I just need to be sure before I buy the Japanese version of Zelda.

It's not universal, so you should try to research Zelda specifically to see what the Japanese version supports. I haven't heard one way or the other.

Conveniently, there's a NeoGAF post by a guy I know on Twitter that covers the languages for the Japanese launch lineup specifically here. For Zelda, you're in luck! That'll run in English. I assume all you have to do to get it to play nice is have an account with the system language set to English or the region set to an English-speaking territory if that option isn't available in-game outright.

From my experience, region-free Nintendo games (first party) will adhere to the system language and as long as English is supported, it'll change the game into English.