MacLane coherence theorem for "monoidal" category without 1

Sorry, I noticed this an hour after posting: Yes, it's true, see Theorem 3.1 in MacLane's excellent paper "Natural associativity and commutativity".

It's a mystery to my why this is so scarcely mentioned.

The proof of Mac Lane's coherence theorem is fairly ad hoc. However, higher dimensional rewriting provides some general methods to prove coherence-like theorems. Using this, it is very easy to prove the coherence of "semigroup categories".

See for example "Coherence in monoidal track categories", by Y. Guiraud and P. Malbos.