It is illegal of an ordained minister of any CHRISTIAN congregation I am aware of to perform a marriage ceremony without a marriage license. AT the very least, his church will lose their tax-exempt status.
I agree with the rest of your post though...
Withing their church rules perhaps. But as a matter of law, I can pay a willing priest to perform the marriage rite and forgoe the certificate recording the marriage. The law only cares about recording licenses. This is why in the USA priests are also notaries in order to execute the marriage license at the same time.
A very comon example is when a priest perform the service for a couple which had to be civily married in advance in order for him or her to process their green card papers.