The state can specify the requirements for a common law marriage and then it is up to the couple to,satisfy those requirements which don’t necessarily include a piece of paper. But if you have a wedding in a church and the statutes don’t prohibit the church from issuing a piece of paper, then that paper could be the functional,equivalent of What we now consider a marriage license. Again, it might take some statutory changes in most states, but I don’t see why it can’t be done. It would probably mean though that more people would blow off getting the paper altogether but they could still claim common law marriage otherwise.
I assume the couple get a piece of paper that shows that they met the requirements.
I assume that if the church is allowed to issue a piece of paper, that the paper would have to be filed with the government, rather than in each individual church or Muslim mosque, or Hindu Temple, and atheist don’t have churches anyway.
Paper will exist, and the purpose of licensing will still exist, there have to be records.