I think you are misunderstanding the real logic of succession to supreme power in the political system built by Augustus and his caesarian-aristocratic party.
It is less the son of the emperor that became his political heir than the heir chosen by the emperor who became his adoptive son.
Of course, this heir was chosen among Augustus' closest relatives, but he was not necessarily the closest one. There were other political or personal reasons which could lead the emperor to chose as heir someone who was not his closest blood relative. Tiberius was not a blood relative of Augustus, and Germanicus was only the great-nephew of Augustus (though married to his granddaughter Agripinna), while Agrippa Posthumus was his last living grandson.
Caligula was only great-nephew to Tiberius who had a grandson.
Nero was stepson, son in law, and by blood a great-nephew to Claudius.
By stating this, I do not mean that Octavien Augustus did not try by all means to have his closest blood relatives as successor. He in fact always did but had to deal with life hasards given the poor health conditions of the ancient world. Many died while babies, children and young adults.
Augustus' first chosen heir was the first husband of his only daughter Julia : his only nephew in the Julian line (because he had an elder stepsister from a first marriage of his father), Gaius Claudius Marcellus, his closest male blood relative.
But this young man died aged 20 and had no child with Julia.
So Augustus married his only daughter to one of his closest friends who was his best general and his most important political support and lieutenant : Marcus Vipsanius Agrippa. And as soon as they had sons, Augustus adopted their first 2 sons as his own sons.
Bad luck, these 2 adoptive sons who were his real grandsons and closest male relatives died, respectively aged 22 and 20.
And given his doubts about Agrippa Posthumus who was adopted at the same time as Tiberius in 4 AD, Augustus had to find a plan B. So he wanted to have as his heir his great-nephew Germanicus, whom he was very fond of, and who was married to his beloved granddaughter Agripinna who gave him many sons and daughters (the ideal roman family and the supposed strong base for a long lasting dynasty).
But Augustus was old, of fragile health, and was careful to have as a heir someone who had been enough tested. So he engineered a double adoption : he forced his recently adopted son Tiberius to adopt Germanicus as his own son and political heir. He thought that Tiberius who was aging would not rule for a long time and that Germanicus who had the support of the core of the caesarian party and was very popular wold then quickly become emperor.
Tiberius lived much longer than Germanicus and was emperor for 23 years ...
So, it is hard to say what would have happened if Augustus and Livia had had a son. In fact they did have (in 37 I think), but this son died shortly after his birth.
And if this son had lived longer, would he have lived long enough to become his father's successor ?
Many people died of natural death when in their late 40's or early 50's at that time. So this blood son of Augustus could well have died before 14 AD.
He could also have died at war like Drusus, the father of Germanicus and emperor Claudius.
Would this son of Augustus and Livia have had children of his own ?
Caesar only had one daughter who reached adult age, and though she was pregnant several times from her husband Pompey, either the children died at birth, or she miscarried before giving birth.
For all these reasons, even if Augustus had had a son, the risk is very high that you have in the end Tiberius succeed to Augustus as emperor in 14 AD.
And if this son ever succeeded his father, I would point to a possible risk of another kind which is interesting.
Augustus seems not to have been very good at breeding children. What the ancient sources show of his 2 grandsons Gaius Caesar and Lucius Caesar, which he adopted as his first 2 sons, and who were raised by him, is that they were more or less failures. Although very popular, they seem not to have been equal to the task Augustus wanted to assign to them. And they seem to have been aware of it and to have felt hemmed in by this awareness.
Gaius Caesar who was the elder of the 2 brothers and the last surviving one (Lucius died in 2 AD), and after being wounded in Asia, he dispaired Augustus by ... asking to be left alone and rest. He died shortly after robably from health complications.
At last, I don't see why this son of Augustus would have to kill Tiberius and Germanicus. Accusations against Livia's poisoning rivals to her son are pure malicious gossip. Many people died young from illness.
Everybody knew since the proscriptions that the roman emperor was and needed to be merciless. After his own daughter who was banned and imprisoned for life because of a plot in 2 BC, Augustus did of course not hesitate to ban and imprison his elder granddaughter (another Julia married to a Lucius Aemilius Paullus) and to have the husband of this granddaughter executed when he discovered that they plotted against his authority.
PS : this second family plot was probably aimed at directing Augustus' plan for succession in another direction than the new one they guessed Augustus considered in 2 AD just at the time Lucius Caesar died, leaving Gaius Caesar as the only heir of Augustus. The conspirators probably knew that Gaius Caesar was fragile, and might have wanted to bring their own satisfactory answer to the question : "what if Gaius Caesar dies young or is not able to become emperor when the old man dies ?"