First thing that comes to mind is the ignition switch. Fiat Ign. switches
tend to be unreliable, so it's well worth checking.

Two recent problems I've met are switch won't turn, WD40 doesn't help,
so with column locked it's a tow job; other was a burnt and charred
contact block on end of switch. Once you get the column shroud
off, a challenge, then a simple 12V bulb in screwdriver with flying
ground clip is all you need to see if all the switches outputs come
on when they should.

What you have working seems to be the non-ignition switched
stuff, start is a momentary contact on its own in the spring
loaded position, so the car starts. It's the supplies that come
on with the ignition that you don't seem to have. These power
things that drain the battery if they didn't go off with the Ign.

With some Fiat Ign. switches you can just change the contact block,
from experience changing the switch body is a pain in the arse.
Tip if you do have to change the switch, don't use the break head
security bolts, use ordinary hex head bolts. No idiot is going
to try and remove the switch to nick the car.