Why do we need to "get the economy moving again"?
Trade creates value.


Previously, Kate preferred spending \$50 on food at Alice's restaurant (rather than cook her own food). And Alice preferred spending \$50 getting her hair cut at Kate's (rather than cut her own hair). That Kate must now cook her own food and Alice cut her own hair means that value has fallen (where value is broadly defined as the degree to which individuals' desires are satisfied).

(Note though that the fall in value in the above example is probably not very large. In most other examples and cases, the fall in value will be much larger—consider for example the children who no longer go to school and are now supposed to be learning at home.) 


Money is not the source of value.


You write:


  It seems to me that all the money that hasn't been spent in restaurants and hairdressers during the COVID-19 pandemic hasn't gone anywhere, it's still sitting in the bank accounts of the people who would otherwise have spent it. 


This expresses a common misconception among non-economists. Money is not the source of value.†

Value is the satisfaction of people's desires—and such value arises through production and consumption. If everyone is sitting at home and not producing anything, then no desires are satisfied, no value is produced, and all the money sitting in banks is worthless.

†The mercantilists of the 17th and 18th centuries likewise believed that gold and silver were the sources of value and that one's country should try to accumulate as much gold and silver as possible. Adam Smith pointed out that this was mistaken--value comes from production and consumption, not precious metals, pieces of paper, or numbers in a bank account.