It honestly doesn't bother me. In fact, I enjoy imagining why/how a Pokemon would know an attack of a different type.
Wigglytuff learning Thunderbolt? It's partially based off a balloon. Static electricity, anyone?
Arbok learning Fire Fang? Well he's got the teeth; maybe he can utilize fire in his body somehow to a small degree.
Azumarill learning Toxic? All Pokemon that can utilize TMs can learn Toxic, which brings up a whole other category of conspiracy theories and creepypastas altogether!
It's also mentioned in various sources that Normal-types, while plain, are still able to utilize moves of many different types, which adds to their uniqueness.
Just because a Pokemon has one type doesn't mean it can't utilize other abilities. You even mentioned a good example, with some Psychic-types who may learn Rock Throw. Pretty much all learned moves make sense if you take into account all options and think about it for a bit; at least that's what I've noticed.