I don't see any running light function on the switch. Where do they connect to?
Assuming the switch connects "t/s main wire" to the signal wires, the wiring would be pretty simple if you ignore the running lights. Your list looks right so something is not connected as per the list, or the switch is not as shown.
The flashing power (from the flasher) should presumably go to "t/s main wire" as you show. Confirm that connecting a lamp from the flasher wire to ground powers and flashes the lamp. To get the normal flash speed you need a two standard lamp load.
If that works then check the switch connects to the signal wires.
You can connect the running lights to switched power if you want them on. The stock switch turns off the running light on the flashing side to make the front signal more visible, your switch doesn't have that function.