If mass/gravity bends space, why does a stationary object START falling in the first place?

Grasping General Relativity at a conceptual level is simply beyond all but a very few. There are mathematical solutions that are already very complex, but beyond that is, really, inaccessible. The trampoline analogy is a very simplistic 'story' that, as you noted, soon breaks down. The curvature of fourspace is not for the faint-hearted. In his time, Einstein thought there were a few others who understood his theory.