This creates an issue with some scholars since they believe in the religion they study.
Not really. Everybody is biased, historical Jesus studies postdate those on Socrates (making Socrates probably the most studied person in history prior to the early 20th century), and most importantly the historical Jesus quest began and continued to be biased
against Christianity. To the extent religious biases exist in historical Jesus studies (and they do), they do in multiple ways. It is widely known, actually, that seminary tends to create atheists, agnostics, and liberal Christians out of fundamentalists as the entire framework of historical inquiry and other methodological tools of the NT scholar, theologian, etc., render problematic fundamentalist/conservative Christian apology.
Irrelevant figures are moot. I was not talking about other gods but historical figures.
So having presupposed the answer, you evaluate the question. It is hardly intellectually defensible to regard evidence counted for the historical Caesar as "moot" when it comes to the historical Herakles or Dionysus yet meaningful when it comes to other deified individuals. You assume certain results from historical inquiry whilst trying to critique vast swathes of said scholarship.
We have coins minted by Caesar and coin made from a month to years after his death.
And we have more evidence for Herakles and other mythic figures. You can't use two standards of evidence by presuming the historicity without rendering the entire enterprise pointless.
People have begun questioning Homer as the author of the work attributed to him due to the lack of material evidence.
What do you mean "have begun!!!??" Even the ancient Greeks questioned whether he existed, and it is practically universally agreed that Homer never existed (although there are those who argue that both the Iliad and the Odyssey were written by a single author). And it has NOTHING to do with the material evidence. Nobody questions whether Euripides existed, whether Socrates did, whether Antiphon did, or whether most of those whom we know of did despite the fact that the vast majority are known through a handful of medieval manuscripts (and most in manuscripts in which they are quoted or referenced, not copies of texts written by them). The reason it is widely believed Pythagoras existed despite the fact that the earliest known biography of his life was written many centuries after his supposed lifetime while 60,000 books on the nature and existence of the historical Jesus were written in the 1800s alone has much to do with the fact that there are no Pythagoreans but many Christians. There is no Reimarus for the Mithraic mystery cult, no Strauß for Herakles, no Wells or Doherty or Carrier or Price for Socrates, etc.
You missed my point. We have material evidence, not just textual, covering part of Alexande'rs life, namely his conquests, which covers a large part of his "world".
You missed mine. Such evidence is all contingent. Historical inquiry isn't about what evidence exists but about how best to explain the evidence that exists, which includes everything from Greek drama to the spatiotemporal distribution of extant 2nd & 3rd century NT manuscripts. Using conquest as evidence, Romulus and Remus were the historical founders of Rome, Agamemnon existed, etc. We don't have material evidence that Alexander, Romulus & Remus, or Agamemnon conquered anything. We have sources indicating this, and after careful consideration any rational person should agree that Alexander clearly DID conquer and therefore existed, while the conquest of the other three occurred but is attributed to mythic figures.
Unless you want to suggest that the collapse of the Persian Empire, hellenization of the area and the emergence of successor states following Greek culture spontaneously appeared with zero cause at all.
Rome existed, and Roman historians not only told us that this was do to the conquests of Romulus and Remus (whom, as Livy explains in a classic instantiation of the historicizing myth, were thought to be raised by wolves because of a misunderstanding of their adopted parents' names), but also that they and therefore the founding of Rome could be directly traced to the survivors of Troy. By your reasoning, they clearly existed. Or we could realize that the historical method is not to assume who is and isn't historical and then singularly apply in a haphazard method particular (and rather
ad hoc) weights to particular types of evidence.