Is Newton's third law a consequence of first and second law?

As pointed out in the comments, your argument cannot be applied to systems with more than 2 interacting bodies. For $N$ bodies, you will arrive at $$\sum_i^N\sum_{j\neq i}^N\mathbf F_{ij}=0$$ which does not specify $\mathbf F_{ij}=-\mathbf F_{ji}$ for all $i,j$ for $N>2$.

Therefore, Newton's third law is not obtained here. It must be an additional assumption.