A based self-map of the punctured disk inducing the identity on the fundamental group is based-homotopic to the identity

Passing to an actual wedge. Combine the deformation retraction with the collapse equivalence of [F1]. They give based maps $a:X\to R$, $b:R\to X$ with $ba\simeq {\mathrm{id}}_X$ and $ab\simeq {\mathrm{id}}_R$ relative to their basepoints. For $g=afb:R\to R$, [F2] and $f_{\ast }=\mathrm{id}$ imply $g_{\ast }=a_{\ast }{f}_{\ast }{b}_{\ast }=a_{\ast }{b}_{\ast }=\mathrm{id}$.

Homotoping the wedge map. Restrict $g$ to each actual circle summand of $R$. Its based-loop class equals that summand's standard generator because $g_{\ast }=\mathrm{id}$. By [F3], it has a based homotopy to that summand's inclusion. The finitely many homotopies agree at the wedge vertex at every time and hence glue continuously on the finite quotient $R\times I$. They give $g\simeq {\mathrm{id}}_R$ relative to the vertex.

Returning to the punctured disk. Compose the based homotopies to obtain $f\simeq bafba=bga\simeq ba\simeq {\mathrm{id}}_X$, all relative to $d$. For $n=0$ the wedge is a point and the same argument is the based contraction of the disk. This is a homotopy of maps on $X$; it claims no extension to any puncture. Only finitely many based-loop homotopies and the specified finite graph equivalences occur, so no choice principle is used.