Primary carers are not just stay at home parents (it’s worth stating it’s perfectly valid to not want to be one just as it is to want to be one). My husband and I have two kids, we both work and I’m listed as primary carer at the daycare (because my position is permanent therefore I get paid leave, my husband is a contractor). The daycare knows each of us equally though. All the tasks you’ve mentioned need to be done regardless of whether you work or not and modern parenting actually does demand more equal roles. Covid has further pushed this. My husband does more cooking, I do more cleaning. We have all the kids activities in a joint digital calendar which we both manage. We both balance being strict parents with being patient and fun. The 1950s model no longer works so if you don’t think you can cope with any of the chore parts of being a parent and discussing an equal parenting relationship with your partner, you may need to rethink it. I apply this equally to men. Don’t have kids if you don’t want to put the time into parenting. I write this having spent the last week caring for two small children with gastro - and my husband has equally been up all night too - projectile vomiting from a baby and a preschooler means no one gets to play at being 1950s husband.
I have 2 children - the youngest is 3.5 months. I think people have a right to child free events if they want and shouldn’t be guilted about it - children can be very disruptive and the style and tone of a function is very different with kids. Just because I find my kids delightful, doesn’t mean they are delightful to others and my friendship with others that warrants my inclusion to an event is not necessarily a friendship with my kids too.