Your symptoms sound very similar to something I had several years ago on a 06 Fiat Grande Punto 1.3 MJ that I bought as a non runner after previous owners garage had fitted lots of parts and ended up damaging the timing chain by violently trying to tow start it, stupid really as with no fuel getting to engine they could have towed it to Scotland and it still wouldn't have started!.
The cause was the trigger wire at the fuse box for the fuel pump relay which also supplies a signal to the starter from the ignition switch.
So if you bridge the contacts at the relay socket the car starts, but even with a new relay it doesn't.
The trigger wire seems to supply a + or - signal to activate or not, the relay.
I had to flip the fuse box over and locate the wire to follow back through the loom across the bulkhead to the ECU.
Having found this I ran a new wire up to the ECU on the opposite inner wing and just where I was going to make the connection I found the original wire had rotted through a couple of mm from the ECU plug connection. When you gently pulled on the wire it appeared to stretch as only the plastic holding it together.
Let us know if helps.
This was about five years ago so I can't remember which contact wire at the ECU , sorry