Well, you don't seem to have anything in $db (and aren't using strict or you would have seen that). But other than that you seem to be just going about it the hard way...

my $datetime = strftime( "%F.%T", localtime );
system( join( ' ',
    'mysqldump -qce --single-transaction',
    $db, @valid_tables, '>',
    "> /backups/table_dumps/$db.$datetime",
) );
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