OP, four years in a village isn’t long, certainly not long enough to be fully integrated, no matter how many roles you’ve taken on. It takes time, quite a lot of time. And not being invited to one acquaintance’s party is in no way an indication of anything other than one individual having 150 people he’s closer to/related to/ reciprocating an invitation to.
Also, you’re doing that thing I often see people who self-describe as introverts on here doing — it’s as though that the effort you see yourself putting in in having volunteered and joined things, when it’s not in your comfort zone, should be ‘repaid’ by reciprocal overtures of friendship from others.
Kindly, it doesn’t work like that.
I spent eight years living in a village doing everything always recommended on here to integrate. I had a child in the baby and toddler group, which I ran for a couple of years before I returned to work FT, then DS was in the village school, I joined the PTA and helped out with village events, volunteered as a litter picker , got involved in a local public transport campaign, went to the pub, invited people over, talked to people, tried to join the local book group (which said no). I’m a socially confident person, and have never struggled with friendships elsewhere, but no progress in eight years contributed to our decision to move on. And people said they were terribly sorry (and very surprised) when we put our house on the market. I think the assumption was that we’d always be there, so there was no rush! And maybe in eight more years we’d have made friends, but I didn’t have eight more in me.
I hope your experience will be better, or that you have more patience than I do!