No, the 64 boxes are only useful for serving as pass-through storage if you want to move a Pokemon from one GB cart to the other and you only have a single transfer pak to load at a time. No matter what format you want to play, you can only bring Pokemon from the GB cart (party or PC boxes there), a predefined list of rental Pokemon that they provide for you, or a team you had already registered from one of those sources. Of course, the Switch Stadium games don't even support connecting GB carts or any equivalent--you're solely limited to rentals there, and you have nowhere to put the "prizes" you earn for things like beating Gym Leader Castle.
