I also have a lot of difficulty seeing what's going on, and am no expert, so take this with a grain of salt.
Here's one small picture (which you might already know) that I found helpful: Massey's description of "vanishing cycles at angle $\theta$" (see https://arxiv.org/abs/math/9908107, around page 23) which gives geometric intuition for the passage from $X_{\infty}$ to its universal cover. Namely, restrict your family to the segment where $t$ is in $e^{i \theta} [0,\epsilon]$, i.e., a ray emanating from the origin in the complex plane; then proceed as you did before, except no crazy covers needed, because your base is now contractible. This gives a functor (isomorphic to nearby cycles for any fixed \theta) together with an action of monodromy.
