Amnesia Brace can help you train pokemons without getting you any experience, it probably would have made sense earlier when it was introduced but I don't think so it is much useful now since most mons can be leveled down to their respective tiers, so there would be nothing to worry even if you over level something while EV training.
Only advantage I could think of it is if you're making LC pokemons and you want to avoid cancelling evolution everytime, you can train them with regular exp share but there's always a chance you might look away and they end up evolving.