To be honest for me the conversation has turned too much about friendship.
There's being friendly vs developing a friendship. At my time so far at the school gates, I'm not sure there's a single Mum I will remain friends with after DC moves on to new schools etc. But in this phase of my life - the parents of the children in DC's class are important.
I suppose I view other Mum's a bit like work colleagues. It would simply not be ok to take instant, or unjustified dislikes to people. It would not be acceptable to act in an excluding manner, blank people or actively try to avoid someone (even if they were pursuing you for a meeting and/or you didn't really like them).
In work mostly people are friendly, inclusive, and collaborative and it's much easier and nicer being that way and we have a lovely office culture. There are on occasion the 'difficult people' but you do your best to navigate them.
This is essentially how I approach parents, teachers, extra curricular parents/instructors and anyone else I need to be in association with while I raise my DC.
You can be open, friendly and inclusive without entertaining a friendship.
You can decline a playdate without leaving one Mum sat on the sidelines at a sports event. You can say no to a coffee, and still move with grace and wave/smile and exchange chit chat at the school gate (even if you've declined several times - you can still do this. You can say - I'm very sorry we just don't have the time at weekends and still turn up perfectly friendly and pleasant at the next event).
You can have a group of Mum friends which you do enjoy spending time, and which you may have a WhatsApp group and you can also see a lone Mum stood at the school gates and you can move to include her in the general waiting for the bell chatter.
That's being socially aware vs socially inept. If people can't switch and flex like this in social places then you are socially inept.