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 essential 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 a simplicial interval and it seems that constructing similar 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.