If an Arcane Shift was countered, does it count in order to draw with Ezariel?

Yes. According to RiftJudge FAQ #9631, RiftJudge FAQ #8979, and RiftJudge FAQ #92, your Ezreal legend ability can still be activated if the spell you played was countered.

Why it works

The requirement for the Ezreal legend ability is satisfied the moment you choose your targets and the spell is finalized on the chain, not when the spell successfully resolves.

1. Finalization: When you play a spell like Arcane Shift, you choose targets as part of the process of finalizing it on the chain. Even if the spell is later countered (e.g., by Defy), the act of choosing the target has already been completed and locked in.
2. Countering: Per Rule 425.1.b, a card that is countered is not considered to have been "played" for effects that require successful resolution (like Viktor, Innovator), but the "choosing/targeting" action has already occurred.
3. Activation: Because the Ezreal legend ability specifically requires that you have "chosen enemy units and/or gear... with spells," the condition is satisfied the moment the spell finalizes on the chain, regardless of whether the spell subsequently resolves or is cleared from the chain by a counter.

The Sequence

1. Playing the Spell: You play Arcane Shift and target an enemy unit/gear.
2. Finalization: As the spell is finalized on the chain, the choice of target is locked in. This satisfies the condition for Ezreal.
3. Reaction: Your opponent plays a counter (like Defy).
4. Resolution: The counter resolves and removes Arcane Shift from the chain.
5. Activation: Since you have already completed the "choosing" action, you may now exhaust your Ezreal legend to activate its ability and draw a card (provided you are in an Open State where the chain is empty and you have focus).