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First of all, it doesn't make sense to only have one, sometimes it's immersion breaking, for example: Why the fuck can't a Gyarados both look intimidating and have moxie? You'd think moxie would make it more intimidating?
Second, most Pokemon already have one ability that clearly outclasses all the others, and this would make it more balanced and interesting because then the weaker abilities wouldn't get shafted, they would still see play and be weaved into strategies.
Second, it's unnecessarily restricting, Pokemon would have alot more tactics and nuance involved, if Pokemon had multiple abilities: Pokemon could play several different roles at once and it would make it easier to counter cancer mons like say, Dracovish, Dragapult, Toxapex, Lando-T Cinderace, etc. without sacrificing team slots for a dedicated counter, like you could bring a Seismitoad with Water Absorb, Swift Swim AND Poison Touch all at once to function well on a rain team, KO Toxapex, Ice Punch Poison Touch to hax and chunk Dragapult and Lando-T, without becoming fodder to Dracovish. Then compliment it with Defiant Inner Focus Pressure Focus Sash Bisharp to gain free Swords Dance from Lando-T Intimidate, and can also abuse rain to tank Pyro Ball, if Cinderace does High Jump Kick then it survives with Focus Sash, counters with Psycho cut and KOs with Sucker Punch. See? More role compression, more variety of ways to deal with threats.