Yes, you're right!
Ref BOAC,
Yes, that is correct, too. However, most of the single-engine procedures I've seen had a diversion point well before the point where you'd be at 1500 feet. As you know, with an engine failure after the diversion point, you're expected to continue on the normal, all-engine departure route.
But, you provide a very, very good point.
PantLoad