First of all, thanks for writing this. I agree that society would benefit from more men working in arts and humanities.
However, I'm not sure prestige is the issue. There are jobs with far less prestige, that pay more particularly relative to the required skillset. Jobs with high pay and low prestige are dominated by men.
While it might be true that a given field becomes more lucrative simply by attracting more men, I feel that causal relationship is reversed. Men tend to gravitate towards professions that are lucrative. Most men do not have the freedom to choose a career without considering how well it pays. Men are generally valued in society by how much money they make.
If we, as a society, want to make professions more gender neutral, we must decouple men from the provider role first.