Lost productivity? Those are non essential people, there was no productivity lost in reality. Are you going to make up something about how they were spinning gold or some such to claim that "no really they are very productive people"? Don't get me wrong, I'm sure every one of those people do the best they can at their jobs, so I don't mean that as a personal insult to them. My claim is that those people are non essential people. They are not required for defending our borders from a massive invasion, they are not required to judge legal matters, they are not required to mitigate our laughable trade imbalance, and they are not required for other members of society to perform their daily activities.
There is of course a red herring where you could claim that a service industry that relies on that many Government workers suffered. It's a false argument of course, because if we took away those non-essential jobs and returned the tax money to those of us that pay, that service industry would make the same amount of money.
Now to the other point you made in "Considering that we have to pay 800,000 people for time they didn't (couldn't) work", this is another line of crap from politicians. We don't "have" to pay them! This was a politician's decision to GIVE them money. Many of them are going to get Unemployment in addition to getting PTO. We didn't have to give them anything, but a politician chose to give them YOUR TAX MONEY! Makes you feel good don't it?
I don't think you understand the difference between non-essential and non-productive. For instance, the FDA was considered non-essential because the country could still function for a while without food inspection (or so they thought, a few thousand people who recently got salmonella might disagree if they could get away from the toilet long enough to post). Systems might need to be upgraded -- anyone working on improvements to existing infrastructure would be considered non-essential. As for the "paid time off" argument.. Well, they didn't exactly ask for this time off, did they? If you were working at a private employer and they said "we can't pay you, and you can go home, but we promise to pay you back at some indeterminate time in the future", would you consider that a paid vacation? I wouldn't. I would consider that time to look for a better employer. I was a federal employee for 3 years, but I left 6 months ago for the private sector because the benefits, pay, and stability of a federal job were terrible. I'll say that again, the benefits, pay, and stability of a federal job is significantly worse than in the private sector.
Complain all you want about government employees, but at the end of the day most of them bust their ass for people like yourself who demand that they all be fired.