This should be the top answer. For one, that gap is too wide for caulk—you won’t be happy with the result no matter how careful you are. Shoe molding is the way, and it’s flexible enough to ride the small ups and downs of the floor. If you want a super smooth look you can do a thin bead of caulk on the top on the shoe molding where it meets the floor trim, but NOT between the shoe molding and the floor because the wall/trim will expand/contract differently than the floor. Good luck!