IIRC, by the old board game rules, cavalry have half the shock and fire in mountains and forest. Man for man, that still makes them better at shock than infantry even in mountains. When the fire phase is first introduced into the game, it is good to have some infantry in battles, otherwise you lose battles. Regardless of whether you win the battle, cavalry are the best at chewing your enemy up. Cavalry are faster; they can retreat from those high attrition zones. I make use of almost all cavalry armies.
Infantry are mostly useful for a small composition in battle to have fire phase activity. They are also useful for assaults and are a cheaper way of maintaining some siege forces. Actually cavalry can usually bounce between sieges, keeping two going at once with two cover forces. So cavalry can be cheaper to maintain in terms of number of sieges in progress.
Where you have seen references to cavalry improving morale is in using cavalry to improve the morale of men coming off of a failed assault. After a failed assault, the infantrymen's morale is broken. Merge the infantry with high morale cavalry, separate the infantry once more, and the infantry are ready to return to the assault with higher morale.
The cavalry are not much use to assaults otherwise, so they can be used as morale boosters for infantry with low morale. Cavalry may similarly be used confidently in an assault to slightly boost morale, but only to just tip the balance when an outcome is about to be had. Troops moving through a province under assault are not part of the assault until stopped.