To worry that my social awkwardness might affect my daughter’s ability to make friends at school?

I’m like you but it has never affected my kids’ friendships. I was not friends with their friends’ parents and they were not friends with my friends’ kids. Mine were old for their year but were doing play dates on their own fairly quickly and most parents are actually happy to facilitate their children’s choices in friendships and are not all as cliquey as MN sometimes suggests.

YANBU, lots of people worry about this I think. Few things I'd say, firstly I doubt you come across to others as socially awkward as you feel inside, and I suspect some of those people who seem as though they're gliding through life as social butterflies are feeling equally awkward or anxious or frantic underneath. School lunches and leavers destinations don't sound that weird as topics to me, school events are always tricky to judge the tone, pure small talk about the weather or whatever is really dull but you hardly want to get deep and meaningful about politics or religion with a bunch of strangers who you have to see at the school gates either. At least you stuck to school-themed things!

Second, yes of course social skills development is hugely important at primary and you're right, IME socialisation at this age does 99% happen through the parents and I'll be honest, it often is quite cliquey with the same 'popular' mums and kids always getting together with one another and not inviting others). But I don't think it hugely matters or is damaging to the kids (don't think they tend to notice when they're this little really unless the parents make it a big deal), they get a lot of practice of their social stuff at school both structured through group or partner work in class and also free play and playtimes. If they do after-school club or holiday childcare plus maybe some classes or other structured activities like brownies, that's ample socialisation for a lot of kids without needing to add lots of playdates and meet-ups too, particularly if dare I say your DD may turn out like her mother to be more on the introverted side and need some downtime and alone time to recharge too.

Don't get me wrong it is nice to do the occasional playdate and particularly as she gets older she will probably ask / want to do more, but you'll work it out as you go and as she finds her feet. I've found with the prevalence of What'sApp and so on you can easily identify from the class group who is who's mum and simply message them and ask even if you aren't particular friends at the gates, in the old days when you had to sidle up and ask in person it was all very mortifying particularly if they clearly wanted to decline, with messaging they can easily just make up some excuse or whatever.

I think the key thing is for you to model social confidence and ease to your child, and also to model healthy friendships, letting her see you around your friends and showing her what good relationships look like. I don't think that making friends with other school parents is necessarily important. I never did the school run at DS's first primary as DH and I were both working and most days he went with his childminder or to breakfast club, so I hardly knew any of the parents. I just invited the children he liked on playdates, on the (comparatively rare) occasions I could do so. He's always been happy and sociable.