[Shazaam!]

The fuel pump is controlled by the engine management computer. When you hit the run switch, the computer powers a relay under the seat that sends power to the fuel pump and turns it on. The computer then checks the crank rotation sensor to see if the engine is running — if not, the computer cuts power to the relay after about five-seconds which then shuts off the pump.

(The fuel pump supplies fuel under pressure as long as the crankshaft is turning, i.e., the pump is switched off by the ECU if the crankshaft stops turning as a safety feature to help avoid fires following an accident.)

Is this initial sequence working correctly?

 

[Shazaam!]

Unplugging one or both relays and put back in the same place, no relay action, no pump, no start.
Unplugging and swapping over relays, all is good again, bike fires up and runs.
Until I switch off for more than 10 seconds again, after that – nothing. I can repeat this as often as I want with the same result. Same with new relays. Also, disconnecting the battery and reconnecting seems to work as well and the bike runs again.

You may not know that the 1.6M controls it's own power supply. That is, when you turn off the ignition, the ECU stays powered for up to 15 seconds while it does some housekeeping.

Try pulling the harness connector off of the ECU and wait a few seconds, then re-plug it and turn the bike on. Does it start?

Also, it may be worthwhile to try a different battery.

 

[Shazaam!]

I don't know much of anything about the ECU's internal diagnostics and initialization sequence. A Magneti Marelli engineer perhaps might know if your symptoms are a result of an ECU component failure.
There's some smart experienced folks over at Ducati.ms - The Ultimate Ducati Forum so you should trying posting in their Technical Section for advice.

That said, the problem on its surface seems illogical from an external circuitry point of view. That leaves the ECU as the suspect. Perhaps your dealer or a friend has a IAW 16M ECU they'll let you try. You don't need to actually run the bike on the borrowed ECU — just see if the startup sequence works. You won't damage the borrowed ECU/Eprom doing this. Good luck and report back.