I'm not so sure about this, though it depends on what you mean by "purely secular". I don't know if there's any data on this, so I can only speak anecdotally - I know people who are sympathetic to religion and believe it can be useful (even one who's a fan of Jordan Peterson, the person who has made a career about talking about religion in a secular way. Aside - I think it's telling that Peterson continually says good things about religion but is still atheist and still doesn't go to church.), but still are within the purely secular realm.
To be sure, I'm not saying that you're necessarily wrong about secular institutions not being able to fill the role that religious institutions can. But at the same time, it doesn't seem like getting people to agree on the benefits of religion gets them to accept the resurrection of Christ, existence of God, or anything like that. Which kind of makes sense, right?
(Another aside - I have always sort of felt like this emphasis on religions being useful over religions being true is a kind of implicit concession to atheism. It's as if religion actually is just a shared delusion, but that's ok, because it's a useful shared delusion. I have no trouble seeing why an atheist would not see this as a compelling case. But the case for religion being true isn't nearly as weak as this "pragmatic delusion"-style argument implies.
Not a criticism of what you're saying, it makes sense to say what you've said in the context of the thread. I'm just thinking out loud.)