Suppose, hypothetically, that 10% of all managers are outright sexists and won't promote women. (This is a thought experiment, I'm not making any claim about the real world.) Then, if equally-qualified men and women apply for the same jobs, you'd expect that the "type of job" class is going to include some higher-paying jobs for men about 10% more frequently than for women. If you "control for" those factors, you are eliminating from your sample all women who work for outright sexists. It might be true that the wage gap is (almost) zero once you do that, but your experimental design is flawed.
Plenty of people have done actual experiments and found that many hiring managers are biased, though I don't think anyone is claiming they are outright sexists. Unconcious bias is plenty bad enough to explain current disparities in opportunity for minority genders and races. (https://www.google.com/search?q=Resumes+with+male+names+are+...)
