I didn't thoroughly read all of the prior posts, mainly the pictures to produce the sketch.
By code the wild leg is required to be identified with orange wire or tape. Since he has undersized wire connected that should be corrected when he wires it permanently. He will likely have more issues once he installs the belts and puts a load on it. Were just discussing how to get the controls working at this point.
Also the code requires that the wild leg be place on L2 in all cases, except for utility meter cans. This is why the factory starter control connections are tapped from L1 and L3, even provides an extra lug hole, only there for the purpose. Its standard procedure.
The fuse is also required for any control conductors that leave the control enclosure, such as the pressure switch and the low oil switch. The supply circuit should be on the order of a #6 or #8 wire, the breaker to provide the circuit protection, is too large to protect #14 control wiring circuits, once they leave the enclosure.
The starter is wired correctly from the factory to comply with code, its the field wiring that leaves something to be desired.
Is this NEC code or local. Is it new? Is this something specific to compressors? I don't ever recall seeing a fuse in a control circuit and I have hand my grubby hands inside quite a few motor controls.