stebulus

The question was whether every σ-algebra which separates points
contains the singletons.  Let's consider (σ-)algebras of clopen
sets of a topological space.  (In fact, by Stone's theorem, there
is no loss of generality in doing so.)  The desired conditions for a
counterexample then correspond to topological conditions on the space:

clopen sets separate points   ::  space is totally disconnected
singletons are not clopen     ::  no points are isolated
clopen sets form a σ-algebra  ::  G_δ sets are open