Scenario 0) I do nothing, get nothing.

Scenario 1) The government gives me $1000 and I buy $1000 dollars worth of goods and services, so now the economy is better by the net gain of $1000 worth of transactions - i.e. the mutual benefits of trade.

Scenario 2) I have a job, do my job and my employer pays me $1000 dollars. Then I spend that on $1000 worth of goods and services. Now the economy gets the benefits of $1000 worth of trade plus everything that I did for my employer, which is in excess of $1000 according to his calculation.

You compared Scenario 1 to Scenario 0 and said that Scenario 1 is a win, but the context was 1 versus 2, where it's a negative. In scenario 2, the employer could be the government paying me to improve the roads, for example, instead of the government handing me the money they would have paid me. In which case the difference is the improved roads.

So the benefit of giving that money must provide a gain greater than the deadweight cost. That gain could be non-monetary (a mom raising her kids for example) -- but just the 'spending' doesn't benefit the economy because spending $500 actually cost much more than $500.