So you think women don't go into plumbing because of cultural norms? Really?
Ok, I wish somebody would study that and find the reasons. The baby argument doesn't hold, though, because it makes a huge difference if it is your own baby.
Better comparison would perhaps be care of the elderly, which involves cleaning up shit but also has a social component.
If the argument there is so few women coders why isn't it also why are there so many male garbage collectors (I could list a great many undesirable jobs here), prisoners, homeless and combat fatalities? Is the goal to balance those out as well?
It is not just cultural norms, the strongest female athletes are about as strong as average men: http://media.townhall.com/_townhall/uploads/2015/11/13/8.jpg (linking to this chart without context because I don't have a lot of time to Google - but if you google, you will find lots of information on the differences in strength).
I doubt that the women/girls who choose to compete in weight lifting hold back because of cultural norms. That would be a silly sport - basically it would be a competition about who cares the least about cultural norms. In any case, there should be several women who don't care about cultural norms (what about the raging feminists who reject the norms, for example).
"women/girls who choose to compete in weight lifting"
That is where the problem lies. When a 6 year old girl tells her parents she wants to become a weight lifter, her parents will say "how about ballet instead?". When a 12 year old girl tells her class mates she is a weight lifter all her friends will laugh at her.
The cultural norms prevent girls to participate. If they are strong-willed and decide to pursue weight lifting despite all of this, they would have a good chance to become very good because of their determination.
But there are female weight lifters. You mean they probably only started later in life, and they would be as strong as men if they started at 6? Not sure if weight lifting is even recommended for kids, what is the average starting age?
Also, I googled and a lot of the best female weight lifters look Chinese. I have a suspicion that they might be recruited by the Chinese system at early age and pushed for the glory of China, at the expense of their health. So cultural norms wouldn't be the issue here.
Women take care of children, typically, and their shit does not smell like roses. So you need a new hypothesis.