In the other hand: All the spell caster's types can learn any spell??
Mages can learn any spell on the spell lists. Priests gain certain spells based on their Path choices.
Mages casting healing spells?? Shamans conjuring undeads? Priest throwing lightning bolts??
Just because you can
learn a spell doesn't mean you have to
Besides, it would be highly appropriate for a Priest of the Thunder God to throw lightning bolts at their enemies.
Even if this is true then a Priest have advantage against a Mage 'cuz the later have a miss chance the priest don't have Don't you think??
What Priests may gain, Mages make up for with a wider range of spells.
Is up to me The GM grant the access to the schools of magic to every class?? Is a way to do it??
Take a look at the Sorcery Campaign Quality. There are some options there to cover exactly what you're after.
And welcome to the forums!