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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

The familiar example of ℚ satisfies the first two properties but not
the third, ultimately because the infimum of countably many positive
distances may not be positive.  With suitable replacements for the
notions of "distance" and "positive", we can arrange that such infima
must be positive and thus obtain a complete counterexample.

Let X be a set and let P be a poset with least element ⊥.
Let d: X×X → P be a P-ultrametric, meaning that:
    for all x in X,     d(x,x) = ⊥;
    for all x,y in X,   d(x,y) = d(y,x); and
    for all x,y,z in X, either d(x,z) ≤ d(x,y) or d(x,z) ≤ d(y,z).

For c in X and r in P, define the ball of radius r centred at c to be
    B(c,r) = {x in X : d(c,x) ≤ r}

Exercise: For any r in P, the balls of radius r partition X.

Let F be a filter on P, that is, a subset of P satisfying:
    for all a,b in P, if a ≤ b then a in F implies b in F; and
    for all a,b in F, there exists c in F with c ≤ a and c ≤ b.
(As Wikipedia says, a filter represents some notion of "big enough";
we'll use it as a substitute for "positive".)  Assume further that
F is nonempty.

Exercise: The balls with radius in F form a base of topology for X.

Exercise: In such a topology, all balls with radius in F are clopen.

Now, the counterexample: let S be any uncountable set; take X to be
the power set of S; take P to also be the power set of S, ordered by
inclusion; take d to be the symmetric difference function; take F to
be the filter of co-countable subsets of S; and generate a topology
on X as above.

Exercise: For every distinct x,y in X, there exists r in F such that
d(x,y) ≰ r; consequently, X is totally disconnected.

Exercise: For every x in X and r in F, there exists y in X with x≠y
and d(x,y) ≤ r; consequently, X has no isolated points.

Exercise: F is countably downwards directed (that is, every countable
collection of elements of F has a lower bound in F); consequently,
G_δ sets in X are open.

This forum question from 2005 has other examples.