Oh, no, it is BY DEFINITION money taken out of the economy. After all, an economy is a system of voluntary exchange, so if you in-voluntarily take money, you're removing it from the economy. That you then put it back in is a different matter.
And no, public employees (of which I am one, BTW) don't contribute to the economy "just like everyone else." Everyone else has to convince people to hand over their cash in return for a service. Everyone else has to prove their value at every transaction. Public employees get to skip that kind of important step.
So the delay: when people are taxed and (just as importantly) threatened with taxation, they don't so much have that money to spend anymore. The resource is removed from them. But the public employee doesn't spend the money the moment it's removed from the citizen; instead the money hits the books of the bureaucracy, then it's eventually transferred to the employee account, then the employee eventually decides to spend it. Each of those steps takes time.
In fact, as I said, even the threat of taxation removes some spending power from the guy about to be taxed. He can't operate based on the assumption that the money will be there two months from now if it's going to be taxed away next month. Therefore, not only is there a delay, but the delay shows up EVEN BEFORE TAXATION OCCURS.
It might be a month between a tax levy and a paycheck increase, but it might be six months between when a business scales back spending in anticipation of the tax, and when the paycheck is issued.