Remove religion to create a church-like experience and you have nothing to hold people together.
Also worth noting that in most churches people have a reason greater than their own selfish interests in socializing to hold them together.
Unfortunately the median age in UU seems to be about 80. I'm not sure 20-somethings looking for community are going to find it among grandparents.
What we really crave is a place for discussing ethics, morals and general life guidelines.
Church has many interesting properties that have been abandoned (due to obsolescence of the content, but not the form!): you can be just a passive observer in Church. You're with other people, which generally act supportively in this environment, and you can just be there amongst others without speaking, only hearing both comforting and hopefully useful ethical guidelines. You're hearing that you're loved and that everyone should love eachother -- which is usually true in the first case, and generally accepted in the latter -- but sometimes a reminder is really due.
Then if you have a problem you don't feel like sharing with anyone else, you have this guy, the priest, which has essentially a confidentiality agreement that you can safely tell him whatever. If you have a problem you can ask for guidance. The priest can usually pull some strings to get other churchgoers to assist you if necessary too (especially easy but important things like emotional support).
What we have today as non-religious individuals are for the most part psychologists, which have many issues:
-- You don't get the participation feel from joining a large group of people, and interacting with the ones you are interested in that group;
-- They tend to view your problems as disorders, when sometimes all you need is the presence of a group of people for alleviating light emotional distress;
-- They don't feel too confident giving advice, and they probably shouldn't, since they may not have experience with your situation (whereas some churchgoers might);
and very important:
-- They are expensive, while church is completely free.
In a web jargon, psychologists don't scale very well. Suppose everybody went to see a psychologist, and each psychologist had 10 patients. Then at least 10% of the population needs to be psychologists. Instead you could get k -- a lot better than 1 -- mutual confidants by connecting with a k person subgroup out of a larger meeting. And when you can't find a suitable subgroup, you have a speaker giving general advice for a very large number of people.
We need non-religious churches.
I've never gone, but I've thought about it.
(I'm not a Christian, so can't say too much.)
If you want something that's certified to be Christian-free, you might want to look up the humanist branches of the Freemasons. Or some of the humanist Quakers.
Or just pick any random hobby like swing dancing for a community around it.
There's a good site for this kind of thing though, if someone just wants to talk something over - https://www.7cups.com/. Not a group thing like church, but a pretty good stop gap until something better comes along I suppose.
In some countries. In other they're free.