Starter clicks using the key, engine starts by jumpering the relay

Hard to keep track of what you have or not done. I tend to "jerry rig" all kinds of nonsense. At this point..........

1...Keep the "path" of the key / start/ relay coil/ NSS path in mind. for the relay to pull in and activate you need three things:
A...A good relay of course
B...A ground through the NSS switch to the relay
C..."Start" signal on the yellow wire at the relay

So maybe obtain two 12V test lamp probes, or MAKE some probes, even small nails--solder scrap wire to small brads or even small sheet rock screws. jam one into each of the two "push on" terminals at the start relay. When twisting the key, the "start" wire should light with power, and the NSS switch should NOT light. If you put the shifter in gear (D, R, etc) both lights should light and it should not engage.

If (1) checks out and or you can hear the light click of the relay, and it won't crank, double/ triple check the starter / solenoid circuit--the big wire from the "square" screw terminal on the relay down to the solenoind. Try once again bypassing that and confirm the starter will crank right after the key did not

Sometimes, even on simple circuits, it helps to draw out that section so you know what all /where all it goes.