>Wouldn't it make sense for me to try & bring more men into the profession, in order to keep salaries low and fill my vacant teaching positions?
I think at this point salary already doesn't matter as wages are already depressed. In other words, fishing for true workplace equity isn't an altruistic endeavor as much as cost savings. Once that is achieved, it's irrelevant who populates the industry.
Nobody is interested in hiring male teachers as costs are already down and there's plenty of female applicants lined up, even though they actually should in the interests of equity and workplace representation.
Your government bureaucracy argument brings up an interesting tangent. Government agencies should be one of the most inclusive workplace environments, (looking at some US administrations, they generally try to espouse that trend [0][1][2]) so it's only reasonable to assume that the same principles would trickle down to heavily regulated industries. If anything, heavy government involvement would mandate such quotas. But they don't. Which leads me to believe there's something else afoot.
[0]https://www.whitehouse.gov/briefing-room/statements-releases...
[1]https://facts.usps.com/postal-service-diversity/
[2]https://www.pewresearch.org/fact-tank/2021/01/28/racial-ethn...
the tinfoil hat stays on and secure