If I pay you and then you pay someone else and then that person pays me the same amount we've increased the GDP without actually doing anything.

Wrong. This transfer of funds would be taxed in some form or the other. GST if u r a registered business. If u aren't, it would come under personal income tax. Therefore, it would not be profitable for you to do this money exchange without expecting something in return. If you say that you simply enjoy seeing cash change hands, then u r generating value. U r getting pleasure in return of doing this experiment. Soooo that does come under GDP.

Now of course, GDP shouldn't be the only metric you judge a country by. It's clearly very flawed in measuring stuff like quality of life per person, wealth inequality and so on. However, it doesn't mean that you should completely ditch it either. It certainly has its use case when trying to understand the economy of a country.