I don't understand why big airports at least (may not be practical for small airports) don't have scales to weigh the planes before take-off ... especially with some of the incidents lately (B1900 in CLT crashing in particular).

The plane in fact may have easily been underweight because the weight of passengers, luggage, and even some cargo I believe (like USPS mail) is assumed and not known exactly, and is estimated somewhat conservatively.