Wow, you sound seriously unpleasant.
When dc1 started school, an experienced mum set up a FB group for our class and it's been so useful - copies of newsletters, reminders of non pupil days etc and a quick way to find lost pe kit or ask for help with queries. She's a lovely, not overly sociable woman who was handing down something she'd appreciated for her older child.
So, when dc2 started school, I set up the group. I've not put myself on a pedestal or made myself extra available for slagging off. I certainly don't want to be centre of attention.
In fact, I am what you describe. I get next to no sleep, have a small baby, a child with medical problems and am generally in a rush to get somewhere so put my head down and get on with things - stopping and chatting only when able or if a genuine friend asks how I am or makes me laugh. Some days, I'll be more rested and chilled and strike up conversation with someone new or who looks lonely and try to be warm and friendly.
Luckily our school doesn't seem to be how yours sounds. We don't have 'obligatory' drinks or go to parties with school parents and we don't have ridiculous expectations of the social aspects of the school run.
Don't indulge in this stuff, nowt good'll come of it.