I'm certainly a layman. But it seems that the predicted strings are supposed to have been caused by the hot-dense early universe cooling off in a particular way

I'm certainly a layman. But it seems that the predicted strings are supposed to have been caused by the hot-dense early universe cooling off in a particular way, rather than anything to do with how the hot-dense state happened. So it's not clear that making the universe cyclic helps you solve the problem. A much simpler solution would be just that the behavior of the early universe somehow prevented them forming or made them much rarer than we thought.