It is not " weird or backwards" to want or to have children; it is literally the purpose of male/female relationships at the deepest of biological levels - and biology remains a driving force in human beings.
Many children are borne out of the love between a man and a woman and the instinctive desire that can arise from that to have a child. I didn't grow up imagining having children at all.....in fact I never saw myself as married and with children, when a child myself. Children have arrived out of relationship - and yes, women do usually bear the weight of most of that for obvious reasons of pregnancy and childbirth, and from breasfeeding and also, quite often, from a desire to be with the child when it very young - if possible.
I'm not sure how doing what your body is designed to do is somehow un-feminist? When younger I really got in to what I would call 'Goddess Feminism'. The feminism that was centred in the experience of being female; in the power of the female body; its cycles, its sexuality and fertility. 'Equality feminism' is all well and good - and obviously women should be paid the same for doing the same work, but at the same time there are sexed differences linked to the above which do often mean that women make different choices - and why should they not be supported in those choices?
I note that lots of young couples now get a dog. There are so many dogs around. People seem to use them as child substitutes. A desire to nurture ( to have a focus outside of the couple itself)does tend to arise out relationships