open my $logfile, '>>', $ENV{HOME} . q{/bin/git-shell-wrapper.log} || die "Could not open log file!\n";

The die expression in the quoted statement can never be executed due to considerations of precedence WRT the  || operator. Workable variations would be
    open(my $logfile, '>>', $ENV{HOME} . q{/bin/git-shell-wrapper.log}) || die "Could not open log file!\n";
(added grouping parentheses) or
    open my $logfile, '>>', $ENV{HOME} . q{/bin/git-shell-wrapper.log} or die "Could not open log file!\n";
(use super-low precedence  or operator vice  || operator (update: this is the current idiomatic usage)). See perlop.