This is correct. My wife and I had the unfortunate experience of having to transport an infant daughter from AL-through TN-to KY for final burial. The law allowed us to transport from Funeral Home to Funeral Home. And yes, we had to have appropriate paperwork. In our case, we were a young couple and had to pay for all of the expenses. This was simply a cost-saving measure.
However, there’s a big difference between a Mother/Father transporting a baby casket and the situation reported in this case.
God bless you. We took our stillborn daughter from NJ to NC - hospital wouldn’t release her body to us but to a funeral home and the funeral home people were actually appalled by the cost of flying her casket commercially so they released her to us with the necessary paperwork. Unfortunately the family is dependent on the openness of the funeral home. And my uncle drove the casket of my great-uncle (and maybe my great-aunt later) from GA to NC.