It's more a matter of precision than offence. Trans men who give birth aside, there are lots of mothers who didn't give birth (adoptive mothers; lesbians whose partners give birth). 'Mothers' simply isn't the same category as 'people who have given birth'.
Plus, if what you're talking about is the experience of pregnancy and giving birth, then it makes sense to focus on that, rather than on the much wider experience of motherhood.
To me, it's more confusing that some people insist on using less precise terms because they find the idea of inclusivity offensive.