If your spouse has to enter the hospital for medical treatment, and you do not have a civil marriage license, then you do not have legal rights to be consulted for your spouse’s care.
If your spouse dies without a will and you don’t have a civil marriage license, your spouse’s belongings will not automatically be transferred to you. Much of it goes to the State.
If you have to go to court, and you don’t have a civil license, you could potentially be compelled to testify agaist your spouse.
etc.
These, and others, are the benefits that come with a civil marriage license.
I think the solution would be simple. If things get so bad that priests are no longer signing marriage licenses for anyone, get married in the church anyway (which is really the marriage that counts anyway) then simply go down to your local courthouse to make it “legal”. Two separate acts in other words.
My wife and I had to do that anyway since we got married in Italy and Italian weddings aren’t recognized as legal marriages here in the US. So we had to go to the courthouse for a seperate “ceremony”. And I certainly don’t regard my marriage any less valid simply because the religious ceremony wasn’t recognized as “legally married” in the US.
It’s the marriage in the eyes of God that counts not the state.