The answer is no, and there are plenty of counterexamples. Note that simplicial sets are not relevent here; one can cook up examples with spaces and take their total singular complexes.
For example, there are high dimensional knots $K: S^n \to S^{n+2}$ (i.e., smooth embeddings with $n > 1$) such that the complement $X = S^{n+2} - K(S^n)$ has $\pi_1(X) = \Bbb Z$. A generator is represented by a map $X \to S^1$ which is a both a $\pi_1$- and a homology isomorphism. This will give examples with the exception of your condition on $\pi_2$.
To get the $\pi_2$ condition on the above consider the subclass of those knots such that $n = 2k+1$ is odd and the universal cover $\tilde X$ is a non-trivial Eilenberg Mac Lane space of dimension $(k+1)$. These are called "simple knots." There is a complete classification of these in terms of a certain bilinear form (the Blanchfield pairing). In the simple knot case, the map $X \to S^1$ is an isomorphism on homotopy up through dimension $k$.