Your e5 move just made me think of something! I now have a guess why he played h6. Maybe black would like to play e5 but after our Qb3 move he was worried about Ng5 attacking our f7 pawn with both his queen and knight! His h6 move obviously stops us from playing Ng5.
If I am right about his reason for h6 (planning e5) then your e5 recommendation would stop that plan. I am not saying that e5 is the best move because both Nd5 centralizing the knight and Nd7 attacking the e5 pawn seem like good options for black. We also need to think about who is helped more by the h1-a8 diagonal being opened up.
However we should compare the of us playing e5 to what happens if he plays e5 locking our d3 and e4 pawns in place and taking total control of the d4 square.
I am sticking with my Ne1 vote for now but I admit e5 is at least interesting.