Why isn't Linear Programming less convoluted? [Soft Question]

Just a quick question. So I'm taking a course in linear optimization, and one of the things that we're going over obviously is the simplex method. I just started the class so I may not be seeing the whole picture here, but I'm wondering why the method is so convoluted. In calculus classes we've gone over optimization problems and from what I understand, it wouldn't be much more work to just do the calculus instead of the simplex algorithm. Anyway, insight would be much appreciated!

Linear optimization can be interpreted as finding the highest points of an $n$-dimensional polyhedron. One of these points is necessarily a vertex, so it doesn't make sense to check anything other than the vertices. So the calculus approach would be just to check all the vertices and see which one is highest, right?