It all depends on the age and how developed with their fundamental movement skills. With handled implements like a baseball bat, hockey stick or lacrosse sticks they all require hand eye coordination. The swinging motion of a bat or stick needs to have higher skills in stability. All of the sports that I listed need to have a higher skill in all three fundamental movement skills locomotive, manipulation, and stability. Another things kids are not aware of their surroundings, so having a bat or stick they are going to swing it around and not be aware of other students.
I think that students in higher grades like 4th grade and up should be allowed to use bats and sticks, but with very strict rules. Also it should to to learn skills and not in a game situation.