You are 100% correct. The strings and the ownership are two independent issues, and one does not impact upon the second.
The strings allow the streets and other public areas in the city to be classified as a private domain as opposed to a public one; however, one may still not carry from one private domain to a second one on Shabbos. The way to resolve this issue is by placing food owned by all the residents of the Eiruv in a location inside the Eiruv, and by this means we can consider all the separate private domains to have a common interest. This common interest remains even if the string breaks.