Trouble understanding the correctness of this Zero-Knowledge proof of posession of a discrete log
This is "Protocol 2" shown on page 207 of the paper (pdf) referenced.

Maybe I should start from "Protocol 1" that requires (I believe) $s_i = x e_i^{-1}$, not $xe_k^{-1}$.

With step 5, it was explained at Theorem 4, "Correctness" part (that is actually proving soundness property). It starts with a claim of Peggy unable to always produce an acceptable answer $s_i$ for both cases of $b_i$ at step 4. However, this answer does not depend on $x$ and only depends on $\{r_k\}$ chosen by Peggy herself at step 1.

My view is, a better (I believe) protocol was presented next year by (almost) the same team.
For practical purposes, Schnorr protocol would be my choice for this setup.
This paper remains the first to introduce a proof for DL setup/problem.