Your last paragraph is the actual value of a unionized strike as a subcontractor.
If your employees strike, you can't fulfill your business obligations, and so you get pressured by the people you have a contract with.

The activity that skirts labor law is individual contractors, who are often indistinguishable from employees except for tax status and are much more often taken advantage of.

A contracting company is just a company agreeing to do business with another, and doing so via it's employees. It's basically identical to a auto parts manufacturer selling parts to a car company. A Ford parts supplier is largely just a middleman for managing the production of parts to keep Ford from having to manage that process itself. Ford can't renegotiate those employees contracts, even though their work is directly to a spec dictated by Ford.