I never heard Peterson or any research claiming this. The claim is that women PREFER certain activities, while men PREFER others. Given a free choice, they're (on average) likely to pick different things to do with their time. That is very different from being better or worse in something, in fact it's a huge misinterpretation.
And this directly brings us to my parent post, because I just proposed what I believe is the main reason contributing to huge and very emotional reaction to these theories (and the hate that Peterson is getting for it, among others). People (women to be exact) feel threatened by it, they feel it takes from them the freedom of choice and puts them into boxes. Which, I argue, is misinterpretation of what statistics claim. Statistics don't say anything about any individual person, and deal with purely fictional middle person.
However, I stand corrected, as your own interpretation of Petersons theories itself is a big proof that the fear of females over these theories is not irrational, as obviously many men will not miss a chance to jump to conclusions like "we are better at this", which is a malignant construct that science never implied nor supported. Being good at something depends on many factors, and personal preferences are only a minor one. Intelligence, discipline, motivation, work habits are all much more important, and AFAIK sex doesn't influence these, in fact some researchers claim that women are better at self-discipline (again on average)