Standard mixed-integer programming formulations for the stable set problem on n-node graphs require n integer variables. We prove that this is almost optimal: We give an explicit family of stable set polytopes of n-node graphs for which every polynomial-size formulation requires Ω(n / log^2 n) integer variables. By a polyhedral reduction we obtain an analogous result for n-item knapsack polytopes. This improves the previously known bounds of Ω(√n / log n) by Cevallos, Weltge & Zenklusen (SODA 2018). To this end, we extend a result of Göös, Jain & Watson (FOCS 2016, SIAM J. Comput. 2018) showing that the linear extension complexity of stable set polytopes of some n-node graphs is 2^(Ω(n / log n)). We show that the same bound holds for ɛ/n-approximations of these polytopes, for some constant ɛ > 0. On this way, we simplify several of their original arguments to obtain a proof that is accessible to a broader community. For instance, our proof does not require reductions to certain CSP problems or the notion of Karchmer-Wigderson games. This is joint work with Jamico Schade and Makrand Sinha.