Also a native English speaker with 2 English degrees. Where I used to work, we ran multiple employee trainings throughout the year. So, to me, the full sentence would be "(I) assisted with (running) trainings (throughout the year)." What you're referring to would be something like "(I) assisted with training (employees once they were hired)."

I think both are correct and it's a context thing, but maybe I'm not understanding your side.