Find your niche.
Gameplay-wise, it probably would, in fact, make a huge difference. You have to realize how weak Ice-types are defensively. And the culprit? It only resists its own type. So if you take the only other type that Psychic-type Pokémon resist apart from itself, you can't expect the likes of Alakazam, who are frail to physical damage, to be too happy about that. And to add more salt to the injury, the thought of making Fighting-type moves super effective would put Psychic-types in an even worse state than Ice-types, because it would only be effective offensively against two types compared to Ice's four, while having the same number of weaknesses as Ice. Admittedly, this isn't too fair of a comparison, since the four types that Ice is weak to are far more common than the three (or if you add Fighting, then four) types that Psychic is weak to, but you get the drill.
Nevertheless, I get where you're coming from. I really think that tons of other stuff in the type chart don't make sense. It's just that they have to balance things out gameplay-wise, like jellicentfan1 said.