battle descripions in ancient historians

[M. Caecilius]

Military background was not necessarily that uncommon, considering you needed to have done at least a token stint in the legions to get into a senatorial career. That means Tacitus and Cassius Dio could be expected to have been tribune, be it only for a year and without seeing real action. Suetonius, of course, managed to get a dispense even after Pliny had managed to secure a tribuneship for him. Livy, I believe, managed to stay a civilian, but his rival, Asinius Pollio, fought in the civil wars (though his work does not survive). Velleius fought under Tiberius in Pannonia and Germany.

I'm less sure about the Greeks. Thucydides certainly fought, as an Athenian strategos (and was exiled for losing against Brasidas), Xenophon wrote the Anabasis as well as his History, so he definitely knew what a battlefield (and a retreat) were like, Polybius was in the army too, fighting for the Achaean League as a Hipparchus...

In short, many ancient writers did have military backgrounds. That they did not bother to write about their personal experiences of battle is strange; on the one hand, maybe they felt they did not need to describe what they and the major part of their audience (likewise senators and equestrians who had done military service as a tribune, praefect of an auxiliary/cavalry cohort, or even legate) already knew, on the other, they may have been placed in safe enough positions not to leave many traumatic experiences they needed to talk about.