They are fine if "refrigerated" up to a year, but that is NOT a canning recipe and should not be used if one plans on storing the jars on the pantry shelf. It would not be safe at all.
You are absolute correct and it is a ongoing problem with many older recipes and people that "can" by the philosophy of "but I've always done it this way". Of course anyone is entitled to do whatever they want behind closed doors, but if you are going to come out of the closet and share with others....you better have current and safe information and recipes.
"Any" recipe written before 1998 is obsolete and should not be followed. And any recipe that does not kill any potential bacteria by either a Boiling Water Bath or a Steam-Pressure Canner should only be stored in the refrigerator or freezer.
The vinegar does not kill bacteria, which is what the canning process does. If this woman never got sick it was because she luckily has not had a bad batch (yet) and/or is immune at some level. Others may not be so lucky. One company reports one woman went 44 years before it nailed her, but it did.
BTW, if I came across recipes with statements like that....I toss in the trash and cross that person off my list to trust.
Support bacteria. It's the only culture some people have.