It's actually more a Henry the VIII thing to require public marriage because he wanted to control the peers and who they married.
In the medieval period, couples would go out in a field, handfast and declare they were married, and it was done. No priest needed.
In my tradition, we do not require a public announcement ahead of time, witnesses are required only because of civil requirements. We Protestants are smart not to get entangled into the rules for legal weddings and leave that to the State. I sign-off as the officiant that the marriage happened and witnesses sign, but there's no requirement for a public announcement by our church.
Frankly, I think the Roman Catholic and other traditions mess up families by insisting on separate religious marriage that can only be undone by their hierarchies. I have married a few Roman Catholics who could not get married in the sanctuary because they were divorced. God can judge who did the right thing, I'm all for letting love win. Including same-sex unions.