Edit: I don't think $S^7$ is associative actually, but the same argument works with any H-monoid that doesn't lift to an $E_1$-monoid
Original answer:
An explicit counterexample: Consider the 7-sphere with its H-space structure in the homotopy category of spaces. Then consider $BS^7$ to be the one-object $h\mathcal{S}_\ast$-enriched category whose unique endomorphism object and compositions are obtained from the H-space structure on $S^7$. Then an argument [1] shows that $BS^7$ cannot arise as the homotopy category of an $\mathcal{S}_\ast$-enriched category, since such a structure must arise from an $E_1$ structure.