Edited to make it right:
The “capping off” map indeed is a fibre bundle map. However, the fibre is not the configuration space you suggest. For, consider the case of $n = 1$. Here the configuration space would be a copy of the surface, and the larger Teichmuller space would have nontrivial fundamental group! This is obviously wrong.
When $n = 1$, what is actually going on is that the fibre over $X$ is a copy of the hyperbolic plane. This plane is the space of points of $X$, equipped with a choice of homotopy class of path back to a (fixed earlier) base point. Thus the fibre over $X$ is the universal cover of $X$.