get past the semantics - client/subcontractor, whatever.
a nurse who works for a nursing home that is 100% funded by medicaid/medicare. is she a government worker? essentially, yes. not direclty, but indirectly her wages come from a government expenditure. cancel medicaid payments, she's on unemployment.
Your example supposes that companies who work for the government carry them as their only clients. If a company relies solely on the government they are essential part of the government. Most companies (even small ones) attempt to have more than one contract so that they are diversified enough to survive. Your argument has merit but for a very small segment of companies. Most defense contractors do more than just government work (especially the larger corporations). Besides, it is not like the government is one large homogeneous client (heck, the DoD isn't a large homogeneous client), there is competition for work inside the beltway.
Cheers,
CSG