Struggling To Find Community

Wanted to post this in the parenting forum but I don't have privileges, so posting here

I'd love to get some opinion on this from other parents who are raising kids away from home.

My husband and I are parents to a 6 year old, our only kid. Both of us parents are boring introverts. Left up to us, we would just curl up with books and indulge in some hikes or a documentary-watching. But these days I'm beginning to wonder, pretty strongly, if this nature of ours is beginning to affect the social life of our only child.

We live in the United States and do not have siblings or extended family and therefore other kids who would be cousins for our daughter and organic play-dates. Our nearest family are our parents who live in India. So my daughter really has nobody to play with outside the school setting.

Although both of us are spiritual, we are not overtly religious, so we don't belong in shloka-chanting groups, ISKCON groups, temple groups and other such strongly-knit communities.

We are not closed to any suggestions, so if there is something here that will benefit us socially, we don't mind getting involved and giving it a shot, but I'd like to understand what are those things ?

One of my friends mentioned that they have a regional gang of friends who all speak the same language. I don't know of any such group.

Another one tells me that every Friday, they have a fixed play-date, the parents meet for dinner, kids hang out consistently, week after week. I was stumped to say the least.

Some others tell us about public school kids hanging out together in the same community and neighborhood. Our public school is not so good, so we are not sending our child to that one just yet.

All 3 are colleagues for me at work, but they all live in far-off cities in 3 different directions.

Every year we organize a fun birthday party for our daughter, inviting all of her class friends and parents, our neighbors, family friends (more like family acquaintances ?), even kids we meet in my kiddo's soccer class and stuff. This is not just for the purpose of making connections but this is just the way we know to do it since we were kids. Most folks show up and promptly disappear too.

Is this just me or has every parent been through this phase of self-doubt and introspection and questioning and feeling like a socially inept failure of a parent ? This especially hits home on holidays, teacher-training days or when it's raining dogs and cats and we can't even step out and play in the park or enjoy our favorite beaches and hikes.

Did you have this problem ? Did you solve it ? How did you solve it ?

Please share your 2c.

Thanks