There are several major reasons for that. One of them is that we actually need two different relations: 1) membership relation, and 2) inheritance (or inclusion in more general case). Unfortunately, we cannot reduce them to one relation. If we do (in prototype-based languages) then normally we will still distinguish the role of this one relation depending on the context.
Mathematically, we have the membership relation '∈' between an element and a set, and we have the subset relation '⊆'. Just as we need both of them, we need both class instances and classes. In other words, if somebody argues that classes are not needed, then it is analogous to the statement that sets are not needed and it is enough to have only membership relation among elements. It is possible to develop such a theory but then we will get an alternative mathematics. Or we will implicitly treat some elements as sets.