I always thought European cavalry were much different than American cavalry, so surprised by your statement. I defer to your superior knowledge, of course, but wasn't European cavalry, especially during the Napoleonic Wars, much more of an offensive force than American cavalry? They also had various types of cavalry for various uses, I thought, whereas American cavalry was much more a scouting force instead of a fighting force. Am I wrong here?
Kinda. Fifty years makes a lot of difference.
Napoleonic cavalry was an arm of maneuver. In theory it existed for recon, for screening retreats and advances, and for occasional tactical charges; in practise a lot of the time it existed mostly as a threat "in being", forcing the enemy to take precautions against cavalry attacks and therefore limit their tactical options. While some cavalry were issued with pistols and carbines, in practise these were such short range weapons that the cavalry were still basically a melee arm.
In terms of simple numbers, most countries had far more light (read: recon and raiding) cavalry during the Napoleonic era than they did heavy (read: trained and equipped for shock charges) cavalry. As such, you could fairly say that with the exception of France (who loved heavy cavalry) and Russia (who were experimenting with mounted infantry) cavalry in the Napoleonic era was mostly a scouting force. This isn't to say that they didn't fight, but their equipment and training shows that fighting was a secondary task.
But that was fifty years before.
By the 1860s, European writers and military theorists were grappling with the obsolescence of melee cavalry. More forward-looking theorists argued that cavalry's main weapon was now the firearm; traditionalists claimed that "the saber will always be the cavalryman's weapon." A few high-profile incidents during the 1866 and 1871 wars were seized on by the traditionalists as proof that their ideas were still valid. However, the main thing theorists of the time were arguing was whether armies simply had
too much cavalry: it was expensive and was increasingly not being used, held back for decisive charges that never came. This was probably a fair criticism, European armies of the 1860s and 1870s had a far higher ratio of cavalry than armies of the American Civil War did; but the discussion often focused on expense as the primary factor, and generals never like it when they get criticised by accountants. Mark Bennett is an excellent English-language historian on the topic.
If you compare American cavalry of the 1860s with European cavalry of the 1810s, then yes, while both have a predominantly scouting role, the latter also has far more of an offensive role. If you compare American cavalry of the 1860s to European cavalry of the 1860s, then the main difference I'd argue you'd see is that Americans are simply more practical. The sentimentality and denial that the European cavalry had in the 1860s - sending people and horses to their deaths rather than admit that breech-loading rifles had changed the battlefield calculus - just wasn't something you got as much in the New World.
Past the 1860s, the situation changes again, since the Americans revert to fighting indigenous peoples and the Europeans revert to fighting colonial wars; and the logistical and technological mismatch of these conflicts end up being very different from the peer conflicts of the mid 19th century.