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.