You're kind of proving the point of the article I linked (and what pretty much every story on this subject, often written by men) has to say, which is that men have an inflated idea of how much they do. Sounds like your BIL does in fact do a lot, but that's also a complete anomoly as far as the societal dyanmic goes. Mowing is once a week. Laundry and dishes are relentless. Childcare and elder care are relentless. Driving and errands - in general that's something that women do as part of their second shift. How often does something heavy need to be lifted? Not everyday....
And, it's not like you're going to believe me or change your mind, so I'm back to trying to extract myself. But first, I'm leaving you with that tally list we talked about.
"What would happen if you asked a thousand moms and dads to tally the chores they do around the house? Working Mother recently did in our Chore Wars report—and the results (surprisingly? unsurprisingly?) were straight out of a 1950s sitcom. He takes out the trash, mows the lawn, does yard work and takes care of the car. She does the rest: cooking, cleaning, grocery shopping, laundry, etc. (Worth noting: men and women are both likely to attend kids’ activities.) "