The WAD/Emulation experience suffers here. Unless you are playing on original hardware with specific homebrew setups or a specific N64 emulator that supports controller plugins, the core "Transfer Pak" functionality is broken. You cannot easily plug your Pokémon Red cartridge into a Wii Remote. This strips the game of its primary identity: it is no longer a companion tool; it is a standalone battle simulator. Without your original team, you are forced to use the sub-par rental Pokémon, significantly increasing the difficulty and frustration.
save files to a Wii WAD, forcing you to use the notoriously weak Rental Pokémon Nintendo World Report Gameplay Overview Rental System Pokemon Stadium Wad
If you want to experience a Pokémon-themed WAD today: The WAD/Emulation experience suffers here
The original Pokémon Stadium relied heavily on the Nintendo 64 Transfer Pak. This allowed players to plug their Game Boy cartridge into the controller to upload their Pokémon, play the Game Boy game on the TV (via the Transfer Pak), and organize boxes. This strips the game of its primary identity:
The game is a series of turn-based battles. You rent teams of Pokémon or transfer your own from a Game Boy cartridge (via the Transfer Pak). You fight through four cups (Pika, Petit, Prime, and Poké Cup) and the Gym Leader Castle.