I'm not sure you should be so hard on management until we find out all the facts here. We too have the same problem with our molded parts especially with the very small ones. Weigh counting is commonly used and we supplement this with a count made on our vision system during the qualifying process.
If management is saying QC is responsible as relates to appropriate methods and suitability of measurement techniques (including counting methods)I tend to agree. Who else in the company should be responsible? The point is more - how do we resolve the count problem? This is certainly a quality issue. When you deal with bulk packaging, there is always a degree of inaccuracy using piece counts. It is compounded the smaller the product gets. To resolve this, we always make sure we have the customers agreement to understate. Most customers can live with extra parts - they complain when you don't send enough. Your degree of overfill has to depend on the product and the degree of comfort required as well as the obvious business considerations. Bottom line is - if the customer has no objection, overpack. It will save money in the long run. If this is impractical, you have no choice but to count (physical, vision system, more accurate piece counter).
Whatever direction you go - good luck.