  We give a new, stronger proof that there are only finitely many
$k$-vertex-critical ($P_5$,~gem)-free graphs for all $k$. Our proof further
refines the structure of these graphs and allows for the implementation of a
simple exhaustive computer search to completely list all $6$- and
$7$-vertex-critical $(P_5$, gem)-free graphs. Our results imply the existence
of polynomial-time certifying algorithms to decide the $k$-colourability of
$(P_5$, gem)-free graphs for all $k$ where the certificate is either a
$k$-colouring or a $(k+1)$-vertex-critical induced subgraph. Our complete lists
for $k\le 7$ allow for the implementation of these algorithms for all $k\le 6$.
