Battles of the late Third Age

[Gwaihir the Windlord]

No medieval commander ever fought in the front unless it was abolutely necessary, that is if he was attacked.

This is not true. Richard the Lionheart? Harold, William of Normandy's foe (admittedly killed by an arrow, but said to have been consistently in the action)? And many others. Kings have often fought in battles, considered right for them to do so and, as you say and exemplify with your reference to William the Conquerer, as a boost to morale and to terrify the enemy. Probably exactly what Faramir's army required.

True though, the Eagles could not have attacked infantry in spear/bayonet-bristling, orginised squares (employed not just by the Spanish) any more than cavalry could in battle. Orcs would probably have trouble holding this sort of discipline, although they were capable of doing so; as you say on the mountainside it would have been impossible.
However at any rate, to form squares the troops must first know of an attacking enemy that instigates the defensive response. Attacking from the air and probably out of the sun deliberately (similar to wartime dogfighting), detection of an Eagle before it was too late would have been difficult. The Orcs were most likely preoccupied, either with fighting land enemies or climbing craggy precipices. The Eagles would have been very deadly against such unsuspecting enemies.

It would be interesting to know how many Eagles were killed in this battle, and in the Battle Before the Gates, but that is not recorded.