This is trivially disprovable. Did grocery stores increase the wages of their clerks once self checkout was a thing, or did they keep wages the same and lay off some of them?
Productivity only affects wages when there’s competition for labor, not when there’s competition for jobs. When there’s competition for jobs productivity gains are captured by the company, not the laborer.
> Do business leaders lobby to reduce public sector wages and benefits?
They absolutely do, yes. I’m genuinely baffled how you’ve never heard this. The Chamber of Commerce crowd is always trying to shrink governments (and their tax bills), and attacking public unions is a big part of this.
That doesn't disprove anything. Productivity is an input to and has an effect on demand, but it's not the only factor. It doesn't need to be the only factor to matter. Lots of things matter. Economics is a complex field.
Do business leaders lobby to reduce public sector wages and benefits? That's not something I've heard before. In any case, whether or not there is competition from the government, if there is a gap between productivity and wages, then there are profits to be had by private companies willing to do the arbitrage. In the case of such a gap, the government doesn't really matter unless they are paying above the productivity rate.