Leave strength.
Just because it has two uses doesnt mean its not as good as the others.
Have you also thought of making strength more physical and tank based?
Like ad more attributes and put the health to Spirit.

Spirit--Mana, Mana regen, Health, Health Regen, Spell damage
Agility--Attack speed, Dodge chance, Accuracy, Critical Chance
Strength--Damage, Critical Damage, Endurance

Perhaps Spirit also gives resistance to fear/status effects? This would give Warriors a good reason to invest in it to keep from losing on that investment of strength .... and many games increase your carrying capacity via your Strength stat, that could be another reason for the other kind of classes to have some investment in it.

Also perhaps you could have max health determined by Strength but health regen determined by Spirit, or maybe both contribute to max health.