It does not matter, if bleed SW is ON or OFF.
As you can see in the bleed schematic below, the duct to the starter is upstream of the PRSOV.
At 737NG you are right with closing signal from the (not closed) start VLV.
But at the 737CL the closing signal comes from the start SW when he is in GRD position.
B737CL
Ingo