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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.

[1] https://ncatlab.org/nlab/show/H-space#spheres