Sure, but they're still constrained by the Courts of Appeal and the Supreme Court. And that bias would show up in the final decision: merely having a letter saying the NLRB will take up the case is multiple steps removed from the political appointment concerns.

Tell that to Boeing: https://www.nytimes.com/2011/12/10/business/labor-board-drop...

I'm pretty sure no one wants to start a dispute almost guaranteed 0-1 before an appeals court that may or may not be sympathetic to your plight.

Even in Boeing's case, which was incredibly high-profile, they decided on an a very expensive settlement rather than fight the case.