On a mechanic level, any worthwhile team runs Shulk because he has OP main character abilities. By seeing the future, the screen slows down and goes grayscale and plays out what the enemy is going to do, before it returns you back to the battle and lets you decide how to react and several of Xenoblade's monster's attacks are color coded as a "heavily recommended" to use Monado Shield, Monado Speed, etc. against certain attacks. Notable ones are:
1) Monado Speed: Buffing an ally's speed to evade an attack
2) Monado Shield: Temporarily negating damage from enemy special attacks (that you see coming from Shulk's futuresight)
3) Monado Purge: Removing enemy buffs
4) Monado Enchant: Enchanting everyone's weapons to deal damage to invincible machine enemies
(also that absurdly strong ability that temporarily reduces all damage by 80-90%)
And various other unremarkable, well-rounded abilities (healing, Xenoblade's Daze/Topple abilities, etc.)