I think the difficulty is that for the vast majority of American history, marriages weren't really performed without any religious connotation. Hell, even legally, from what I understand, there was a time where marriages _had_ to be performed by a clergyman of some kind. Now, you're right, the religious association is technically optional, but it's worth remembering that for a very, very long time, marriage was as much a religious agreement as it was contractual.
There was once a time (and still some places) when religion played a role in regulating other aspects of society, like dietary regulations. Nobody says "keep religion out of my kitchen" when the local health department insists that you don't cross contaminate other foods with utensils used on raw meat.
Just because we used to regulate these ideas with religion in the past doesn't mean that our current institutions are religious. Some people who are legally married also subscribe to religious meanings of marriage. Some people who have food handlers cards also follow religious food customs. They're related but independent concepts in the modern day. The modern legal construct of marriage in the US is secular.
Got married in the '00s in the US and IIRC in that state it was still the case that, to be valid, you had to have an ordained minister or certain officials (a judge) sign the paper.
Now, would anyone ever check? Nah. Unless litigation (divorce, inheritance, whatever) came up and someone thought invalidating the original marriage might somehow help their case, though even then, dunno if it'd really matter.
