Of course, the other aspect to take into account is culture & tradition:
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Anglo-Saxons, unlike the Continental German tribes such as the Franks and the Goths, do not appear to have regularly fought on horseback. The poem about the battle of Maldon in AD 991 makes it clear that many of the warriors had ridden to the battlefield but then dismounted and fought on foot. Indeed Byrhtnoth, the leader of the Saxons, gave orders to drive away the horses so that there would be no thought of retreat. Only his own horse was kept so that he could use it to ride around the battlefield. When Byrhtnoth was killed and a large part of the army routed one of the fugitives took his horse to speed his escape. Another account, albeit even later, is the Anglo-Saxon Chronicle entry for 1055. This records how the Herefordshire fyrd, in battle with the Welsh, fled before a single spear was thrown because they had been ordered to fight on horseback, contrary to their custom (Anglos contra morem in equis pugnare jussit). There are occasional references in Anglo-Saxon histories to mounted forces, such as the Northumbrian king Ecgfrith's expedition of horsemen (equitatui) sent against the Picts which is mentioned in the biography of Bishop Wilfrid. However, these do not make clear whether the force fought mounted or dismounted. 'Equitates' is the general Latin term for mounted soldiers as opposed to the word 'ala' which is used only of cavalry. Underwood: Anglo-Saxon Weapons and Warfare
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We don't know whether 'Elvish' battle tactics in all periods/situations included fighting on horseback - Anglo-Saxon warriors could have learned mounted combat (like the Rohirrim), but that simply 'wasn't their style'. It may be that at the time of the Last Alliance the Elves didn't fight on horseback - & probably if they didn't use saddles they wouldn't have - Shadowfax may have been able to keep Gandalf on his back in any circumstances but I get the sense that Shadowfax is a pretty unique horse. A bareback cavalry charge is a recipe for disaster. In fact, mounted combat of any kind is incredibly difficult without saddle & stirrups. And let's not ignore the possibility that the Elves may not have wanted horses on the battlefield.