All attempts to do this, particularly in the internet age, have failed spectacularly. They just turn into the worst parts of organized religion without the higher power. Remember the Reddit atheist movement of 10-15 years ago? A not-insignificant portion of that population went on to become the alt-right.
You're making an observation about the USA type of atheist here, and I'll leave that to you because I understand that in the USA atheists are an unusual thing. But I just want to point out that in the UK and many parts of Europe, being atheist is just very normal. Its so normal that there's not really a label or a scene, and there's also a lot of just agnostic/dont-really-ever-think-about-it type people that would probably be classed as atheist in the USA, but in the UK they're just <unlabelled> because no-one cares whether they believe in deities of some kind or not. And many people might say they are christian when asked (46%), but dont ever pray, or go to church. A lot of UK people have christian weddings and funerals, but thats about the sum total of their involvement. Only 8% of the UK are getting baptised, only about 5% go to church regularly.
For example I see that in the USA its very important for politicians to talk about their faith, and if a politician is atheist it gets talked about as a 'thing'. In the UK its the exact opposite, the last thing we expect politicians to talk about is their faith in god, and if they do talk about it we get kindof weirded out (ref: Tony Blair in the late 90s).
I just want to point that out, because I know the UK exported Dawkins and that might give people the wrong impression. Most of the UK are quiet-atheist or quiet-agnostic. And they are quiet because its no big deal, there's no political or moral stakes, no-one gets judged for being non-religious over here. (Even in politics, where you might expect political opponents to nit-pick everything about each other, no politician here would ever call out an opponent for being atheist, because it would just be a nonsensical thing to call out, and they would be laughed at for even trying to call it out)
Offer not valid for Christopher Hitchens, Douglas Adams, Ricky Gervais, Percy Bysshe Shelley, Bertrand Russell, and Salman Rushdie
Cycling clubs and board games groups can bring people together, but they don't ask for the same kind of commitment or engender the same sense of shared identity.
It isn't that they cannot substitute and create many of the benefits, but are unlikely to.
Humans are emotional, physical, and (whether you interpret it as supernatural or not) spiritual creatures. The ritual and belief in something bigger than oneself are an essential part of the process. If you look at other organizations that serve similar roles to conventional churches (such as the Grangers, Freemasons, Shriners, The Lions Club, etc.) they all replicate the ritual and higher purpose commonly associated with churches.
Church is way more meaningful than that. Baptisms, deaths, marriages are run by churches and the meaning can survive the majority of attendants not showing up.
The problem in modern society, is teaching the meaning behind churches. It takes longer than a couple cycling classes and is more demanding on the spirit.
What powers the typical football club "factotum", showing up every week to cut the grass, inflate balls, wash shirts, clean changing rooms, etc etc...? What powers the cyclist agitating about the need for bike lanes, or proselitizing the sport? There is no seriously rational explanation. It's a type of faith.
There is a lot of appeal to some practices of the mountain Mennonite/amish in CO where I'm at.
But a lot of those things that appeal to me, and trying to organize something like that, could quickly turn into a prepper cult ;0
Full disclosure: I'm an atheist UU and a member of my church's board of trustees.