I'll let Plutarch answer this one.

For if, as Iphicrates analyzed the matter, the light-armed troops are like the hands, the cavalry like the feet, the line of hoplites itself like chest and breastplate, and the general like the head, then he, in taking undue risks and being over bold, would seem to neglect not himself, but all, inasmuch as their safety depends on him, and their destruction too. Therefore Callicratidas, although otherwise he was a great man, did not make a good answer to the seer who begged him to be careful, since the sacrificial omens foretold his death; ‘Sparta,’ said he, ‘does not depend upon one man.’

For when fighting, or sailing, or marching under orders, Callicratidas was ‘one man’; but as general, he comprised in himself the strength and power of all, so that he was not ‘one man,’ when such numbers perished with him. Better was the speech of old Antigonus as he was about to fight a sea-fight off Andros, and someone told him that the enemy's ships were far more numerous than his: ‘But what of myself,’ said he, ‘how many ships wilt thou count me?’ implying that the worth of the commander is a great thing, as it is in fact, when allied with experience and valour, and his first duty is to save the one who saves everything else.

Therefore Timotheus was right when Chares was once showing the Athenians some wounds he had received, and his shield pierced by a spear, in saying: ‘But I, how greatly ashamed I was, at the siege of Samos, because a bolt fell near me; I thought I was behaving more like an impetuous youth than like a general in command of so large a force.’

For where the whole issue is greatly furthered by the general's exposing himself to danger, there he must employ hand and body unsparingly, ignoring those who say that a good general should die, if not of old age, at least in old age; but where the advantage to be derived from his success is small, and the whole cause perishes with him if he fails, no one demands that a general should risk his life in fighting like a common soldier.

Such is the preface I have thought fit to make for the Lives of Pelopidas and Marcellus, great men who rashly fell in battle.

The problem is a little more complicated than you make it seem. In a lot of ancient armies, the general's role was to give the right example. Whether he liked fighting or not was immaterial; he had to fight, or else his men would not respect and follow him. This kind of inspiring bravado became especially toxic because of Alexander the Great, whose contempt for death set a precedent for every Hellenistic king to cultivate a reputation for bravery in order to be regarded as fit to rule.

Needless to say, there are lots of stories of generals dying in battle, and in many cases their cause died with them. The Persians lost the decisive Battle of Plataia when their commander Mardonios was killed. The Spartans lost at Leuktra when their king Kleombrotos was mortally wounded. Yet these men knew they had little choice; if they had held back from the fray, their armies would have crumbled even faster.

The Plutarch quote above shows the tension between the two recognised roles of the general. On the one hand, he had to manage the battle; on the other hand, he had to lead by example. It was difficult to strike a balance. The first general who is known to have habitually stayed out of the fighting in order to command his troops is Pyrrhus of Epirus (he of the Pyrrhic victory).