You still need to have a marriage registered with the state, because that makes sure you and your spouse are entitled to all the rights that come from many different laws that have been in place for married people, long before any of this current stuff of today came around. Not doing that will make it so that you don’t get any of those legal benefits that the many laws have allowed in the past.
One for example (just for one, as there are hundreds) is that death benefits from Social Security won’t carry over to the spouse, without a marriage being seen as legal by the state. All that does, by you not getting your marriage recognized legally, is that you take money away from your spouse. Who wants to do that with their spouse?
But no clergyman needs to be involved with that.
The “don’t get the license” was for the minister.