Does blocking an attack with the Shield spell still trigger Armor of Agathys?

I am contemplating a Oath of Conquest Paladin / Sorcerer multiclass character, with a 1 level dip in hexblade warlock.

This character will have access to the Armor of Agathys and Shield spells, which potentially allows for a pretty nice combo.

The Shield spell states (PHB 275):

Casting Time: 1 reaction, which you take when you are hit by an attack or targeted by the magic missile spell…

Until the start of your next turn, you have a +5 bonus to AC, including against the triggering attack,

It looks like you are indeed hit by an attack, but since you have a bonus to AC, you can potentially force that attack to miss instead.

Armor of Agathys states (PHB 215),

If a creature hits you with a melee attack while you have these hit points, the creature takes 5 cold damage.

If I cast Shield and cause an attack that "hit" me to miss, does Armor of Agathys still deal damage?

I am leaning toward yes, though this combo seems a little too strong–Shield is already a strong spell, and this character will eventually have 9 1st level spell slots and a 7th level spell slot. Not only can they freely block attacks with Shield, but they can also deal up to 35 (5d10) damage to the attacker without an attack roll or saving throw. While a Warlock would have to spend both of their spell slots to do this once, my character would be able to do this 4 times with spell slots to spare.

Some notes:

  • Related questions ask about reducing the damage from a confirmed hit to zero. This question is asking about what happens when the Shield spell causes the attack to miss entirely.
  • A number of online resources, such as the Roll20 5th edition compendium, seem to be missing the correct casting time for the Shield spell. Answerers should make sure they’re looking at the correct version from the PHB.