The late Pope himself discounted the KGB angle.
There is another theory that is not much mentioned.
In the 60s and 70s the Italian Mafia was making a fortune smuggling heroin from Turkey (using an alliance with Turkish criminals with links to the Turkish military junta) and the Bulgarian "KGB" (I don't remember what the Bulgarian version of the KGB was actually called) were partners in the smuggling in a smuggling link that was truly world wide.
The mafia was using the Vatican banking system to launder their drug trade money. The Pope was a threat to that arrangement and the Mafia contracted a hit on the Pope using their Turkish-Bulgarian contacts. Remember the P2 lodge scandal? The banker hanging from London bridge?
I discount the KGB because if they got cought it would have been bad and they would not have used such a hit system. A guy on the street with a hand gun?
Also, the Turkish gunman started to speak about the Fatima prophesy - sounds like someone in the Vatican had him mention this so the Ope can get the hidden meaning of it - i.e. its an inside the Vatican job - you are warned - that is Mafia kind of stuff.
Also, there were claims the hit team from Turkey hid out in a Freemasonic P2 lodge that the mafia also used.
Anyway, after the failed attempt the Italian state for the first time since Mussolini went after the Mafia in a systematic way to break them and they probably did destroy their organization to a large degree.