I think though there is an element of truth in getting stuck in.
For example we have friends who moved rurally, about three hours away from where they previously lived, with no contacts and never having lived there before, to a small village, last year.
They got stuck in, for example a young neighbouring couple let their garden over grow and in passing they mentioned they were struggling, so they said no probs, want us to help, and they said god yes if you’re sure, so we were all invited up and spent a day helping clear, our Male friend then spent about a further week helping them out. The couple also got stuck in and provided drinks and food, they were delighted,
An elderly neighbouring lady was taken into hospital, so they went and cut her grass for her, so she didn’t come back to an over grown mess, and planted some pretty flowers, put her bins out.
They invited all their neighbours round for drinks, impromptu after the pub type events, and some more planned, but they often had a house full.
When the pub had events on they cooked and brought food, went along, joined it, and they helped out, and they joined the local committees to organise events.
When folks in the village were selling things, plants, eggs etc, they stopped and bought them.
When events were held where attendance was a concern, they invited us all up, we bought tickets and turned up, boosting attendance and making everyone happy,
They even pet sat some animals. They literally threw themselves into it, and expected nothing in return, they did the inviting, they did the helping, and they asked for nothing back.
The upshot is they are now invited to everything, everyone stops to talk to them, even in the pub people are constantly coming up to chat.
And when they chat, they ask questions and listen, they make it about the other person “how is your mum, can I do something” that kind of thing.
In a year they have become a key and accepted part of the community. So I think there is an element of it being a two way street, it’s not just expecting to be invited and turning up, but also being proactive in doing the inviting yourself, and going out of your way to help.