The US does have a cavalry tradition - the American Army still fields cavalry units (both air cavalry and cavalry mounted in armored vehicles). Many of these units draw their origins from the wars on the Great Plains that took place in the middle to later 19th century. These troopers were decisive in these series of wars - the distances they took place over were far too large for infantry units to fight in, and most (if not all) Plains tribes were mounted and could easily avoid pursuit by infantry units.

These troopers did not fit in the same tradition you're referring to largely because they did not fight like European cavalry - namely, charging with spears or swords to fight other cavalry units or break enemy formations. American cavalry generally dismounted in battles, especially as time went on and especially during the Civil War. A large part of this was simply timing - while the US was in several wars between the Revolution and the Civil War, they saw few of the set piece battles that would require cavalry. By the time of the Civil War, rifling and tactical changes had begun to make the cavalry charge obsolete, though that wasn't completely the case. However, much more useful in the Civil War was highly mobile mounted infantry, often in the Union's case armed with repeating rifles (especially later in the war).

Another factor was the lack of nobility in the US. Most of the 19th century's European armies were raised from the peasantry, with noble officers, but cavalry often drew more heavily from the nobility. This was for obvious reasons - nobles were more likely to know how to ride, and they inherited some of the knightly tradition in many countries (this is a big simplification, but it generally held across Europe). In the US, there was no nobility, hence not really a tradition of distinctive units for the higher class. Moreover, being able to ride and keep a horse wasn't as distinctive in the US as in Europe, hence it simply wasn't the marker of status that it might have been elsewhere.