I would say that this interpretation goes too far. We ejected aspects of common law which were in conflict with our founding principles, but we retained by inertia much of English society and it's methodology.
I think having the Church of England as part of the government gave marriages within it official government sanction. I think states eventually moved to restore this government sanction by creating marriage licenses. I think society wanted it back.
It doesn't matter what society wants - the AUTHORITY to conduct marriage came from the Church of England, not the government. The King combined both, but the authority was still with the Church because marriage is a pledge before God. When the government split from the Church, the authority to conduct marriage split with it. I know what it claims, but as long as God is a part of marriage, government has no legitimate authority over it. The most government can do is acknowledge a civil contractual union for bureaucratic and administrative purposes. But that is not marriage before God.