I've definitely counted paper the same way. Basically needed to sort short pieces of paper by the thousands. We weighed something like 20-25 sheets then used that weight as a measurement.
If you need a perfect count, then you're correct about the accuracy, but generally a few off here and there isn't that big a deal. Many companies will allow for some error because it isn't worth the time to track it down to perfection. This applies even to food standards: the FDA allows up to 60 insect parts per 100g of chocolate (coffee, the cutoff is "Average 10% or more by count are insect-infested or insect-damaged"). They also allow mold up to a certain %. 4% for coffee, and I'm seeing some say 10% for certain fruits. You can see lists here: https://www.fda.gov/food/current-good-manufacturing-practices-cgmps-food-and-dietary-supplements/food-defect-levels-handbook
Perfection is expensive, cheat a little. Your boss may have been annoying, but in general he's more correct than you were.
Sure, but a big business doing large volume would care less. They generally already order with the built in assumption that even if the amount is correct, not every single one would be usable. At certain costs/products this may require accurate counts (like say docking stations) but with other certain things, including some foodstuffs and of course much cheaper supplies (like say disposable straws or chopsticks) they wouldn't even bother to count to make sure they got 10000 straws this order instead of 9995 straws. The amount of money paying someone to coun that to be sure would be more than the missing straws worth, unless you suspected your supplier was shorting you on purpose.
If you need more specifics, then generally the smart thing to do is find a machine that already counts more accurately than a human, like change/bill counters, or other counting machines. Generally isn't worth it to have any employee count large numbers regularly.