**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 is a copy of the hyperbolic plane.