Hi, just to clarify - two actually did have biological children to take the throne. If by child you mean a male child:

Tiberius had a son Drusus with his first wife Vipsania, and Claudius had a son Britannicus with his wife Messalina.

For both they did not take the throne because of political machinations rather than because of any external factors.

Drusus, according to the sources, was murdered by his wife Livilla. According to Tacitus, Livilla was seduced by the head of the Praetorian guard. His ambition was to succeed Tiberius as heir, but of course Drusus was in the way of this. Sejanus seduced Livilla and then convinced her to poison Drusus, with the long term goal that Sejanus would take Livilla as a wife who would still become empress were Sejanus to take power.

In the case of Claudius, his son Britannicus was overlooked because of the schemes of Agrippina the Younger. She wanted her own son, Domitius Ahenobarbus to take the throne, so she slowly diminished Britannicus’ power at court by removing his advisors, belittling him in the public eye by dressing him up as a child, giving out gifts to the soldiers in Domitius’ name to earn their loyalty, and finally convincing Claudius’ freedmen to advise Claudius to adopt Domitius as his son, which he did and Ahenobarbus took on his other more famous name: Nero. Agrippina then allegedly conjured up a “feud” between Nero and Britannicus, claiming Britannicus was jealous of Nero’s adoption, and sewed a rift between Britannicus and Claudius that made him choose Nero as his heir. According to both Tacitus and Suetonius Claudius later regretted this choice and promised to make amends (I.e reinstall his biological son Britannicus as heir) but Agrippina poisoned him before this could happen to ensure the accession of Nero. We are told by Tacitus that some did question why Nero had been chosen over Britannicus.

But you’re slightly right in the sense that there were succession issues among the Julio-Claudians which made choosing a successor a contentious issue. One need only look at Augustus who, because he only had a daughter Julia, first adopted his nephew Marcellus. Then when he died he picked his stepsons Tiberius and Drusus, then when Julia had children with Agrippa he favoured his grandsons Gaius and Lucius, then when they both died as well as Drusus had to go back to Tiberius as his choice for heir. Most of these heirs died of natural causes or accident, although Tacitus says Augustus’ wife Livia may have had a hand in Gaius or Lucius’ death this is just heresay.

So there were plenty of potential male heirs from among the extended family for the emperors to choose from, as well as biological sons in the case of two of the Julio-Claudians. But political scheming as well as natural deaths led to the choices being narrowed.