Is activating the list working towards Ezreal Prodigal Explorer draw mechanic?

No, activating The List does not count toward the activation requirement for your Ezreal legend ability.

According to RiftJudge FAQ #11001:

• Ezreal’s legend ability specifically requires that you have chosen enemy units and/or gear twice this turn with spells or unit abilities.
• The List is classified as a gear card (Rule 150).
• Because there is a formal distinction in Riftbound between unit abilities (Rule 145) and gear abilities (Rule 150), an activated ability originating from a piece of gear does not satisfy the requirement for "unit abilities."

Therefore, even though The List allows you to choose targets when you use its ability, it does not contribute to the "chosen twice" requirement for your Ezreal legend.