If you live in Australia and are affected by some of the themes within this post, please seriously contact Lifeline to seek help or find the local equivalent organisation in your area to help you.
John Wick needed to be the one to end Gianna D'Antonio's life, otherwise the debt was not paid.
Yes, that's a technicality, Gianna was dying after having slashed her own wrists and John could have just left her to bleed out. But that wouldn't have been fulfilling the "Marker" and in that world, they live by their rules and technicalities. Santino would have just found another "job" for John to have to complete.
John wanted the debt ended, so when Gianna slashed her own wrists, he needed to ensure that she was killed by his bullet specifically. In that case, even if Santino had claimed that she had tried to commit suicide then John would easily prove that he was the one who ended her life.
Santino was obviously paranoid about the whole event, which is why he ensured that his people kill John after the deed was done. Given that, Santino would have suspected that John had ample time to talk to Gianna. However, in this way, John could claim that he immediately shot Gianna as soon as she tried to commit suicide. Whereas leaving her to bleed out would have alerted Santino to the fact that her and John had a nice conversation before her death.