They work to bring both sides together. They attempt to do that by bringing both sides within what they call the zone of reasonableness. That is determined by your standing amongst your peers. That is why short duration and on time contracts are important. They drag everyone up with you. That’s also why I cringe when I see posts complaining that another airline might leapfrog us. That’s exactly what you want! The hammer they carry is the ability to either release pilots to strike or allow management to impose a contract. Normally that is preceded by a PEB or presidential emergency board. The mediators recommendations carry a lot of weight with the PEB. It is possible although I don’t remember it ever happening in my career at any airline for Congress to impose a contract and deny the right to strike. A congressionally imposed contract would likely mirror the PEB recommendations.
The most important goal of the NMB is however to prevent a interruption to commerce. Everything else is secondary.