My ds started school two years ago in a rural village we had just moved to where I knew no one (and everyone knew everyone else). Going to the school gates and parties etc made me feel really anxious every day. I get what you mean. I also don't have many friends, so the stakes felt quite high at first.
My advice is, do you. Just linger, being yourself, and friendly in passing. Things like this come with time. I don't have great mum friends at the school (which is fine), but I do have a few 'mates' I tend to stand with or say Hi to when I see them. Our kids tend to get invited to the same stuff and we have, once or twice, bumped into each other in the local pub and had a drink together. Now the kids are a bit older they come round for play dates and their parent might pop in for a cuppa or might not. Some days I'll pass a mum I know quite well and we'll just nod. Or maybe even not notice each other because life is busy. But that's all OK too.
Don't take it personally. Be available but don't try too hard. Eventually people will just get used to seeing you and say hello. And then one day the school run / kids parties will seem like an everyday inconvenience with the occasional social interaction rather than the "running the mum gauntlet" that it seems like at the beginning.
Dont forget the Covid has set everyone back a bit in making these connections, so everyone is in is in similar position. If I went to a party with no one I knew I'd say to another parent "Crikey, it's been so long since I've been somewhere with this many people I don't know. Bloody covid! It feels pretty weird." Someone is bound to be thinking the same.
FWIW I think parents at school gates are incredibly cliquey (probably for of all of the above reasons). I chose not to try to get 'into' any groups and just he my slightly bonkers, obviously an outsider self. It helped me in the long run. The interesting people got curious, and the mums I do now know are pretty like minded.