Typical symptom (on most cars and motorbikes) of crank sensor. They fail when hot but work OK when cool. However, I am not a V6 expert so perhaps someone will agree or perhaps laugh at this suggestion.
Yes... and no
What it looks like is two faults combining...
A possible crank position sensor failing but..... The give away is the lack of starter motor operation.
But I wouldn't automatically assume it is just that at first, do some simple preventative measures to eliminate other possibilities before spending money.
Next logical issue is a contact fault...
1. Main lead to starter motor, loose or bad connection
2. Main battery leads insecure
3. Bad earth lead to ECU and inner wing
4. Bad contact to the spade connectors to the fuel pump relay....
I would personally start by removing the battery, charging it up and while it's out, remove the relays one at a time and clean the contact spades.
Disconnect all the earth points and clean them all up one at a time...
Then disconnect, clean with electrical contact cleaner and reconnect all the multi-plugs one at a time... It will only do some good as it's routine to do so and might just sort out the issue before spending on parts...
Just to eliminate one other possibility... Once it's running, get the battery and alternator charge state tested for FREE with a "drop test"... Most tyre and exhaust centre's offer this service FOC with no obligation
ALL Alfa's don't like low voltage issues when it comes to batteries and can throw up really wierd behaviour when starting up.
If it still does it after all that is done, then replace the CPS