Why did money replace the barter system?

This is one of those cases where the definition of a particular Word, in this case "barter", is crucial.

If barter means the direct exchange of goods and services without a medium of exchange, the more common use, then by and large, the invention of money supplemented the barter system by providing a nonperishable medium of exchange. Money provided an effective way to avoid the problem of one party being unable to provide a good or service that the other party wanted.

If barter simply refers to exchanging, while many people think the bartering system ended when money was invented, people can still barter and pay with money. Money gives a nonperishable item to be bartered with.