To my knowledge no one has ever done any kind of quantitative analysis on where men were most likely to die in battle, but it's utterly beyond dispute that your chances of dying in battle were much lower than your chances of dying as a result of a camp disease. The question remains, though, why would anyone want to do that?

Psychology is a weird thing, and so is sociology. Peer group pressure among men has always been a factor in their public decisions, and a mix of rational, psychological, economic, rhetorical, philosophic, and moral decisions produce men who were willing to die, or risk death, in wars. The decision wasn't always the same and those factors could be a great deal different even between men serving in the same conflict, to say nothing of different conflicts in different times.

I wrote about this question in an earlier post about Napoleonic column attacks, the psychological superstructure of armies and how masculine cultures contributed to motivation, and why men joined early modern armies at all.

I'd be happy to answer follow ups after you've read through some of those older posts.

This is excellence; this is the best human prize, and the fairest for a young man to win. This is a common benefit for the state and all the people, when a man with firm stance among the front ranks never ceases to hold his ground (...) This is a man good in war; he quickly turns the waves of enemy spears, and stems the tide of battle with his will.

-- Tyrtaios 12.13-22

In Archaic Greece (c. 750-500 BC), battle was not yet about the clash of well-ordered hoplite phalanxes. Instead, the battle lines of early Greek armies were fluid, with javelin men and archers mingling with the heavy infantry, and no one observing any order of ranks and files. In these chaotic battles, warriors were free to choose whether to get right in front and face the enemy, or hang back beyond missile range for safety. No doubt even the well-armoured elite warriors generally preferred the latter. So they had to be encouraged, and reminded of their ideals - to prove themselves, to win glory by personal courage and skill. In the Iliad, Agamemnon roams around the Greek lines before battle, chastising every lord he sees hanging back, and ordering them to earn their reputations by going forward. Above is Tyrtaios, famed Spartan poet, reminding the Spartans of his day (c. 650 BC) that victory could only go to those who went to the front and stayed there. Here he is in another poem, trying a different argument:

It is a fine thing for a brave man to die when he has fallen among the front ranks while fighting for his homeland (...) It is a shameful thing for an older man to fall in the front ranks with the young men behind him.

-- Tyrtaios 10.1-2, 21-22

Here's Kallinos (fr. 1):

Come, let a man go forward, brandishing his spear and mustering a stout heart behind his shield, as soon as battle has come. (...) All the people miss a stout-hearted man when he dies, and while he lives he is the equal of demigods. For in the eyes of the people he is like a tower; he single-handedly does the deeds of many.

Poems like these were sung before battle for centuries after, and they reminded Greek warriors of the heroic ideal recorded by Homer. In these poems, and in the epics themselves, great heroes were given a special title in battle: they were promachoi, "front-fighters". They were expected to go out in front of the rabble, pick out a suitably prestigious opponent, and take him down in single combat. The risks were far greater for these front-fighters, but they were the only ones who gained the glory and the spoils. Here's Mimnermos (fr. 14) celebrating an Ionian hero who faced the dreaded Lydian cavalry:

Never did Pallas Athena find fault with his heart's fierce strength, when he sped forward among the front-fighters in the midst of bloody battle, defying the enemy's bitter spears.

The association with Athena is no accident. She was not just any goddess of war; she was also worshipped specifically as Athena Promachos, literally "Athena who fights in front". A sculpture of front-fighting Athena stood on the Athenian Akropolis and she was depicted in this form on the prize amphorae of the Great Panathenaic festival, a reminder to all athletes and spectators about proper behaviour in a life-or-death contest.

In short, in response to the knowledge that fighting in front was dangerous, the Greeks created an entire battle culture idealising the front-fighter and encouraging their citizen warriors in every possible way to live up to that ideal. Everyone knew it was more honourable to stand in the front rank than to hang back; everyone knew it was the job of every warrior to go forward and never shirk from the fight. Even after battle had become a more organised affair of regular formations, this ethos remained strong, as demonstrated by the speech Lysias wrote for the Athenian Mantitheos to defend his service record in the Corinthian War:

Then after that, gentlemen, there was the expedition to Corinth; and everyone knew beforehand that it would be a dangerous business. Some were trying to shirk their duty, but I contrived to have myself posted in the front rank for our battle with the enemy. Our tribe had the worst fortune, and suffered the heaviest losses in the ranks: I retired from the field later than the fine fellow of Steiria who has been reproaching everybody with cowardice.

-- Lysias 16.15

Even in his day, the early fourth century BC, Greek warriors still boasted about fighting in front; they still boasted that all their scars were on the front of their bodies; some of them would still try to rush ahead of the formation to prove their courage in sight of everyone.

At the same time, all of these sources also show the gap between ideal and reality. There would be no need to encourage Greek heroes to go forward if they would do so of their own accord. There would be no way for Mantitheos to spin his choice to fight in front as an act of exceptional courage if it weren't true, as Lysias makes him say, that many would try to arrange the opposite. The Greeks knew perfectly well how much danger they incurred from fighting in the front ranks (even if modern movies hugely overestimate that danger). This was a big reason why, in Greek warfare, generals usually led from the front, giving their troops the right example. It was also a reason why Greek battle formations tended to be very deep, with rank upon rank of uncommitted warriors ready to block the path of retreat from the front. Any number of methods, from the invocation of ancient ideals to the use of physical force, were used to make the men willing to stand in front of the formation and be the first at the enemy.