Dragoons would've been considered heavy cavalry or line cavalry as opposed to the more traditional role of mounted Muskets/Rifles by the 19th century, but certainly never would've worn Cuirasses in most European armies.

As for the preponderance of single or double breastplates well past the more traditional usage of them; the answer is simply because it still made sense to. Cuirassiers were meant to quite literally bludgeon their way through an enemy line; smash it to pieces. While a perfect charge means the enemy breaks before or at the moment of contact, a brief and violent melee may ensue, and of course braced musketeers pose a significant threat to the life of both horse and rider. A breastplate would do little against a musketball, and were mainly meant to stop slashing attacks (such as a light cavalry saber), spent balls and shrapnel and lessen the impact of a thrusting attack (e.g: A bayonet or heavy cavalry sabre). It also added weight to a charge, and by tradition curiassers were big men on bigger horses, not meant to conduct 'hell for leather' charges but rather choreographed, short and powerful charges at a decisive moment. One also must remember that even concentrated musket fire would be of questionable impact on cavalry gaiting into trot in say, 1815. Until the advent of standard issue, breech loading rifles, it was more sensible to form a square and cease fire with small arms while heavy cavalry closed distance for a charge. Once you could successfully interdict and cause casaulties at beyond a few hundred yards range, the cumbersome cuirasses became questionable; and even then some nations had them for battlefield purposes as late as 1915.

The intelligence of using cuirasses proved itself time and again during the Napoleonic wars; French Carabiners were eventually given breastplates due to horrendous causalities being taken in cavalry scrums; a testament to the perceived life-saving measure of wearing a breastplate. Likewise, Austrian Cuirassers still wore a single breastplate to protect their front.

Not all European armies continued to use breastplates, it must be noted. Prussian Cuirassers were so in name and tradition only, and by the time of the Napoleonic wars no longer used any breastplates whatsoever. Nor does heavy cavalry require armor of some sort to remain effective; the designation of 'heavy' is more a reference by this time to horse, rider (both expectedly large and powerful) and their perceived role and endurance on the battlefield. Many Heavy Cavalry regiments had no semblance of armor; Dutch Carabiners, Prussian Garde D'Corps, the British Household Cavalry and French Grenadiers a Cheval, to name a famous few.