The influence still counts. The influence from yagn's is not provided via trait, ability, or bullet bonus, so it still counts the influence bonus.
Action • Cost 0
Shootout: Choose a dude. Boot that dude and their attached cards. That dude gets –1 bullets and cannot use their abilities. Their attached cards lose all traits, abilities, and bullet bonuses.
Yagn's Mechanical Skeleton
Goods • Bonus 2 • Influence 1 • Cost 2
Weapon • Attire • Horse • Gadget • Difficulty 9
This dude gains the Gadget keyword and has +3 value.
Opposing cards cannot boot or move this dude.