I find myself jealous of these people. For the price of a little hypocrisy, you get to be part of a very supportive community. You get structure for your life, a large number of social gatherings, and a big pool of potential friends and romantic partners. I almost considered going to church just for that reason (back when I was in college and friends with a lot of devout Christians), but I couldn't stomach the hypocr…
Someone did start an atheist church along these lines: http://sundayassembly.com/ This is at least the second time that people have done so: https://en.wikipedia.org/wiki/Ethical_movement I don't have any experience with either movement, so I don't know whether they would be very similar to or very different from what you would like.
My impression is that some of these initiatives emulate the 'wrong' parts of church. While I'm not certain of this, I suspect that this has something to do with the fact that many churches in more recent times have made the same mistake, and these initiatives often involve people with a background in these kinds of churches. My background is Evangelical, and in my opinion they do 'community' extremely well. Perhaps they're the best 'form' to emulate, and since the 'content' is often relatively... flimsy, it is easily replaced with other things.
That said, the initiatives I've experienced were still very worthwhile and enjoyable!
Let me elaborate. Via meetup.com a friend of mine organized meetings every two weeks. Rather than follow the 'Sunday service' model, it was closer to the 'weekday small-group' model that is part of most churches, and particularly important Evangelical ones (to the point of using the Sunday as little more than evangelism).
We'd meet in a group of around 8-14 people and after some initial conversation, we'd split up into groups with a piece of text chosen by the conversation leader. This could be a bible passage, but also something else entirely. Then, we'd start talking about the topic, following a few basic rules. Most crucially, we were strongly encouraged to not primarily debate or discuss our viewpoints, but to try and keep things personal and focused on experience. This, of course, created bonds through the (forced) openness and the resulting recognition and empathy.
Afterwards, we'd go to a bar and just mingle and talk about whatever. Usually much of that would be spending more time talking about the topic we'd discussed earlier.
Many people came because they were tired of 'empty' conversations in the typical social events, often because they had a desire to discuss 'higher' things with 'smart people'. Some ended up disappointed that they had little room to espouse their views and belittle the 'superficial people' with an in-group, but most came to appreciate and value the personal, meaningful conversations. And, of course, inevitably the 'intellectuals' would end up discussing philosophy or religion and the like anyways, but now with a foundation of empathy and friendship.
It was also extremely valuable to see a mixed group of individuals that would often not associate or even avoid each other. One of my favorite regulars was a guy who owned a printing-press in a neighboring city and just came by because his adopted 16-year old daughter wanted to attend. He had a lot of trouble with the 'level' of conversation initially, and clearly was not used to talking about intellectual things or sharing personal-ish stuff. But over time he came to enjoy the thoughtful conversation and the free-form 'pub time' afterwards and many of us, in turn, came to appreciate that he was not just some 'simpleton' and often contributed immensely to the conversation by virtue of having a very different, more experiential outlook on life (and a very different past).
Some of the most valuable things that I had as a church member and that I always missed were the community, the personal interaction within such groups, the encouragement to show at least some vulnerability, and the mix of wildly different people exchanging views and experiences. And those were the best things that these 'sunday assembly' type initiatives offered, in my opinion.