Seems to me that what provides the "meaning and purpose" is group activities with voluntary participation and mutual interaction.

Yup, this is one of three sources of meaning and purpose. Self-reflection and serving others are two other sources.

I imagine that in a non-religious and non-communist setting, it would be the non-profit organizations providing platforms for group activities.

That's true in many cases. In other cases, where there are powerful social democratic governments, research shows that governments provide such venues.

So it depends on how each each society is organized. The key is that every society that survives and thrives does so because it provides its members with means of experiencing meaning and purpose. The structure is less important than the three components that research shows go into meaning and purpose.