"Even in the long run, it's really really hard to cut nominal wages..."
Don't get it wrong. Inflation is a tool to cut wages. It is fundamentally anti-labor.
It is difficult to paint the alternative of artificially restricting the money supply to a level where the private sector [as a whole] must reduce some employees' nominal wages or fire them every time it offers pay rises to its most in-demand staff as more pro-labour. It doesn't sound any more pro-labour when people preferring that arrangement argue that recessions are a more appropriate mechanism to hold down wages, and acknowledge the purpose of zero inflation [and acceptance of economic downturns] is to allow wealth to be preserved for years or even generations without the need for it to be used in job creation.
But it's a blunt instrument. Providing tools to aid those workers adversely affected by those market shifts, whether by declining real wages or lost jobs, is the role of fiscal, not monetary policy.
Because it's not. It's the cost of fiscal policy decisions made overtly to aid “job creators” in the supposed hope that the wealth they drink in will trickle down as a golden shower for the rest of society.