chdir($startdir) should be outside the loop, shouldn't it?

However, I'd look at a rewrite using File::Find or File::Find::Rule. The latter especially simplifies the task:

use File::Find::Rule;

my $start = shift || '.';

for my $oldname ( File::Find::Rule->file->in( $start ) ) {
    ( my $newname = $oldname ) =~ s/\..*?$//;
    rename $oldname, "$newname.xml"
        or die "...";
}
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