[A1971Blazer]

If it starts by jumping at the solenoid then it's not the starter.

You mentioned that there is no voltage to the purple wire, are you checking it at the starter? The purple wire is the hot wire that engages the starter, it passes through the neutral safety switch on the column near the firewall. unplug it there and jump across the connector to see if that fixes it. The starting circuit is independent of the LS harness, or should be in most cases.

The fact that your fuel pump doesn't come on,, also indicates that there is no voltage going to the relay that triggers the fuel pump.

If it's an LS then you should get a 12v signal to the ignition relay as well as the fuel pump relay when you turn the key to the run position. If not there are a couple of fuses or relays in your swap harness that could be faulty.
Did you do the swap yourself?