The Dragon Lord
On the whole I would not want Pokemon to become like the early Digimon or Monster Rancher games in this regard where you have to invest a lot of time an energy in training and rearing your monster to the point where it is the main part of the game, but I like Pokemonamie/Refresh.
I'd say the opposite all around, myself. I have little to no interest in Amie/Refresh/Partner Play as an optional side mode. I just see it as a waste of my time that I could be spending doing actually valuable things in the game. But, if the games where to integrate these features in a meaningful way, and make raising your Pokémon just as important as training it, I would enjoy it as a mechanic. (Disclaimer, I have no idea how Digimon and Monster Rancher works, so maybe I'm wrong.)