Weighing coins was always the preferred way of counting money. The pound sterling is called the pound sterling because 240 mediaeval pennies minted from sterling silver weighed one pound.
And some modern coins are designed so that bags of mixed coins can be counted by weight. For instance, all US cupronickel clad coins (dime, quarter, Kennedy half-dollar and Eisenhower dollar) have the same ratio of weight to value, such that a pound of any combination of them is worth $20.
This is sort of interesting as historical trivia, but at what point would it become convenient or useful? You'd need to sort your coins before weighing them, and it seems like if you were partly sorting your coins you may as well totally sort them.
> such that a pound of any combination of them is worth $20
Also most of the time you can just dump a bag of coins on the table and visually quickly determine an odd ones, remove them, weight the remainder. Sorting != counting.
