An explicit proof can be found in Cole's _Many Homotopy Categories Are Homotopy Categories_ (Lemma 3.4). Some topological properties of the interval are used and it seems crucial to the argument. These properties are essentially that $I \times I$ retracts onto both $I \times \{ 0 \}$ and $I \times \{ 0 \} \cup \{ 0, 1 \} \times I$ and that $(I \times I, I \times \{ 0 \})$ and $(I \times I, I \times \{ 0 \} \cup \{ 0, 1 \} \times I)$ are homeomorphic as pairs of spaces.

This doesn't hold e.g. for the simplicial interval and it seems that constructing Hurewicz type model structures on simplicial categories is much more subtle. Some theory for pretty general enrichments is developed in _Strong Cofibrations and Fibrations in Enriched Categories_ by Schwänzl and Vogt.