3 ideas
Suspicion attaches to the living, so a killer who appears to die early is removed from the suspect pool at exactly the moment scrutiny is sharpest. Wargrave's staged shooting works because the witnesses accept one confirmation of death from a doctor he has already recruited. A verification that rests on one compromised observer counts as no verification.
Once the guests accept that the killer must be one of them, every alliance becomes a risk and cooperation falls apart faster than the murders alone could explain. Each person's logical move is to trust no one, and that general distrust leaves each of them isolated and easier to kill. Paranoia that is rational for the individual becomes a weapon for the one person who actually knows the plan.
The novel sets up crimes the courts could not reach, such as a governess letting a child swim out to drown or a general sending a rival to his death, and treats them as still culpable. It then shows private justice, carried out by a man who appoints himself judge, becoming its own atrocity. Wargrave's confession gives two motives at once: a lust to kill and a genuine obsession with punishing the guilty.