That socialisation telling you that you need to oversee all child related things is also what makes you feel responsible for it
When you've been a single parent for many years, like many of us have, and I certainly have ( from a very young age)........Or when you are quite a strong person who is also staying home with the children....through either choice or circumstance..then that tends to be the way it goes. You are the one that takes on certain kind of responsibilities.
That's not to say if there is a father around he does nothing. Of course that's not the case......and depending on the man, he may actually take over child care responsibilities once home, or certainly do lots of play activities with the children.....may have more patience for long, detailed games, constructing lego..... do the bath time routine, read the bed time stories; and may also do household maintenance of the exterior kind ( clearing out gutters, maintaining cars, heavy lifting type stuff, washing dishes).
Personally, I've always made the decisions about home furnishings, style, decoration, where to go on holiday etc - because I have strong likes and preferences, and these sorts of things matter to me. Of course I pass them by my partner, and we discuss........but really that is all I'm doing. Getting my decisions rubber stamped.
I'm a strong minded person with strong likes and needs, and I always make sure I have time and space to myself, and to do what I want to do. Having creative control of my own life and over the things that I value and find important is key. Some things I'm more flexible about, and I've been really quite happy with doing the cooking, present buying, making social arrangements, play dates, clothes buying......
Some women aren't......but some women are, and are by nature quite comfortable with domesticity. Like being busy all day, like shopping, like cooking.....
We're all different, and different relationships, and the people within those relationships, have different patterns and preferences.
I don't think it is necessarily liberating or personally empowering to have to go out to work full time, if you'd rather not, and/or are in a position not to have to.
I do think it is inevitable that women are more directly impacted by having children. For example if a relationship breaks down and it involves small children...then I imagine very few women who would be happy for the father to have full custody; even though they might be happier with shared arrangements when the children are older.
I don't think any of our choices are always completely free...whatever our circumstances. Life, and events, often makes decisions for you - to a greater or lesser extent