The mental load sounds like it could be largely resolved by GTD, since eliminating mental load is the reason GTD was invented.
And if it's a shared system (eg whiteboard in kitchen) then the man/breadwinner can see at a glance what needs done and pick up tasks without having to ask.
Of course, there's a much more elegant solution, but I will leave it as an exercise for the reader :)
The difference is I don't ask her to do it, and I don't broadcast what I'm doing to end up on a list. It needs to be done so I do it. If you keep tally you will not have a good time either way.
On the other hand stuff I don't need or want moved (my work stuff in my closed home office) always gets "ordered" according to her standards so I have my doubts about whether much of that behavior is society's fault or some people's innate wish to see some things done a certain way or in a certain order. It has to be done based on an internal list of needs or ideals, not because society says my stuff has to be ordered by her standards.
When you're substantially better at something just do it. Your partner will do the stuff they're better at. The rest gets split or shared with a simple prior arrangement. If the result is what matters don't get stuck on the how. And if something doesn't need to be done, don't do it. You'll expend mental and physical energy for no good reason.
The difference in my household is that I, the husband, not only do all the things that you're talking about but also provide "Boss as a service" for my wife. It puts a ridiculous amount of load on me and burns me out. The difference as you correctly pointed out is men tend to "man up" and just do it.
At one point an ex-partner of mine felt like she did way more than half the household workload. I felt under-appreciated because I was basically the only one to "fix" things if something needed to be done in the house.
We decided to set up a simple task manager for ad-hoc tasks, with some alternating recurring tasks like doing the washing. This worked quite well, as it gave some objective facts to balance our feelings and allowed us both to easily pick up our half of the work whenever we had some time.