I think you mis-state the counter-argument, it’s not that (a) there is only sparse-to-nonexistent contemporary evidence of any historical figure that far back, it’s that (b) there is only sparse-to-nonexistent contemporary evidence of comparable historical figures that far back. Lower class Judean figures from the 1st century do not leave much trace on the historical record, even leaders of messianic movements.

The question you ask, though, is a good one. Its answer is dependent upon a different question: evidence for the existence of whom? Who we are talking about greatly dictates what evidence there is, and what kinds of evidence there are.

If we take someone as all-pervasively dominant as Augustus, you have (a) extensive literary documentation in terms of Roman historical writings, like Suetonius, Tacitus, etc.. You have extensive epigraphic evidence, not least including the ‘Deeds of the Divine Augustus’ (Res Gestae Divi Augusti) – the funeral inscription for Augustus found throughout the empire and listing his life and accomplishments.

You have coinage; you have archaeological evidence corroborating significant elements of his life and deeds; etc. etc. That’s why we don’t doubt the existence of Augustus. But this is all the kinds of evidence you would expect to find for a person of such importance at that time.

I’m not planning to drift into a discussion about historicity of Jesus, but Jesus’ impact in his own life is minimal, which is why it’s important not to import an expectation about evidence based on his later importance in Christianity. That’s anachronism.

Pontius Pilate provides another good illustration. Pilate is attested by (a) the Pilate stone inscription, (b) mentions in Tacitus, Philo, Josephus; (c) mentions in the four canonical gospels and a few other non-canonical texts. That’s not a huge amount of evidence. But it is enough that Pilate’s historicity is not generally questioned. Tacitus, Philo, and Josephus all attest to his existence. Jospehus is clearly well-informed about the time period in Judea and local politics, and is generally considered reliable. Tacitus mistakenly calls Pilate a procurator, when he actually appears to be a prefect, but this appears to reflect the fact that by the time Tacitus is writing he would be thought of and termed a procurator anyway. The Pilate Stone, an inscription dated to 26-37 CE provides epigraphical evidence attesting to Pilate and in accordance with the literary depiction of his life.

Overall there is not a huge numerical amount of evidence for Pilate, but enough that exists that it seems petulant to argue against his historicity without valid counter-evidence.

I suppose the question circles around to this: historical evidence for whom? That determines what there is and what there might be expected to survive.