If government workers actually spent all day working, we would need about half the number of them. This would cause mass lay-offs from the public sector, flooding the jobs market with unemployed people. This would cause a spiral of wage deflation, reduced spending power, declining tax receipts, spending cuts, leading to civil unrest, and ultimately the collapse of government and social order.
Let them have their cat videos.
To be fair I work in the private sector and I can't see much difference between your example and all of my work places.