Dragoons were extremely useful and versatile units, so much in fact they were sometimes considered a branch on their own, contemporary 17th century imperial army lists for example refer to 'horse, foot and dragoons'. Earlier in the period they were indeed mounted infantry, but with time they developed into a sort of cavalry/infantry hybrid able to perform as both given the circumstances.

Not every person mounted on a horse is a cavalryman, fighting on horse requires training and skill, and light cavalry in particular tended to be more than just an unit designation. The best light cavalry was usually recruited from warrior societies on the fringes and frontiers of states, and were highly skilled soldiers that relied on individual initiative, mobility and courage, and were mounted on sturdier smaller horses able to travel long distances on little fodder. Mounted infantry is in contrast a much easier way to achieve mostly the same result, as they are not trained in horsemanship but merely use their horses as transportation. They can still engage in skirmishing, raiding and pillaging, foraging, harrassing, scouting and whatnot, with the small caveat that they dismount to fight, and as such they could be useful to fulfill the role of light cavalry without any actually trained light cavalry, which was rather useful as western Europe did not have much of a light cavalry tradition, indeed during the middle ages the English relied a lot on mounted longbowmen and continental forces used mounted crossbowmen, while "true" light cavalry was reserved in places like Iberia due to moorish influence and Eastern Europe due to constant contact with the steppe peoples. Light cavalry would only become popular in western Europe in the 18th century with the creation of Hussar regiments, originally of Hungarian origin, but even then dragoons would not fall out of use because they were still easier to train and could still be useful in the battlefield as fast moving infantry.

Dragoons were rather popular in the age of gunpowder as one of the several tactical innovations of the age that combined firearms and cavalry. It was really almost by design that dragoons had to be mounted infantry, because the usage of firearms on horseback was very problematic, reloading was even more of a hassle than it was already when mounted on a horse, accuracy likewise was difficult to achieve and recoil more problematic, and of course the horse in itself could get scared with the bangs and the smoke, not to mention that it could get burned as the soldier had to handle a lit match cord. A more effective usage of firearms on horseback was achieved with wheellock pistols and carbines, as they could be preloaded and holstered and many of them could be carried. Of course pistols had a shorter range and as such they were used almost like a melee weapon, while dragoons were fitted with carbines, not as poweful as proper infantry muskets but less cumbersome for their purposes. There were cavalry units meant to fire from horseback, employing the so called caracole tactic that was popular in the 16th century, that consisted on ranks of cavalry succescively discharging their pistols or harquebuses and wheeling back around in order to wear down the enemy formation, however the caracole would fall out of use due to various factors, first it had rather limited range, also it was very difficult to pull off and soldiers would often shoot at too far and inneffective ranges and flee to the back of the formation, afraid to actually commit to a charge, then there was the issue of infantry employing more and more firearms and being able to outgun the cavalry, and lastly this tactic was vulnerable to a cavalry charge that cut to the chase with cold steel. While that tactic withered away the dragoon did manage to stand the test of time, as infantry was much better able to employ firearms

So there's the issue that you suppose that mounted infantry could either be used for skirmishing or for rapid battlefield movement, but the versatiliy lays in the fact that they were indeed used for both of those things. Dragoons were a jack of all trades, able to function as cavalry (either in or outside of battle) or as infantry whether the situation demanded, so they could move alongside cavalry and hold positions alongside infantry, although by the same token they were neither proper cavalry nor proper infantry so often inferior in comparison to the real deal. Furthermore beyond just tactical mobility, there's also the larger issue of strategic mobility, which is not getting somewhere in the battlefield quickly, but getting somewhere within the theatre of operations in itself quickly. This can be useful fo several things such as securing a position, surprising the enemy force, reinforcing units far away. Strategic mobility was of paramount importance in places were large expanses of land had to be traversed just to get to where the fighting was going on, for which mounted infantry could get faster than normal infantry. For this reason most of the regular Russian cavalry of the 18th century were dragoon units. Similarly the mighty army of the Persian conqueror Nadir Shah had almost all of its infantry mounted on horses, which made them able to keep up with the operational realities of central asian warfare in which cavalry was essential

Now somebody better correct me if i'm wrong, but i don't think i have ever come accross a single instance of soldiers being carried to the frontline on wagons. Wagons were used for carrying supplies along with the army, but not for carrying the army itself. The only thing i can think of that resembles that idea is the ancient practice of riding to battle on chariots and dismounting to fight, but i don't know much of it to comment on that. Anyways it seems rather inefficient to use wagons pulled by horses to mobilize humans when those humans can move by themselves already, the amount of horses needed to make that wagon significantly faster than the walking formation would be indeed rather great, and as such would greatly add to the expenses of mobilization for proportionally little gain, not to mention it would be inconvenienced by uneven terrain even more than cavalry, which is incidentally one of the reasons that chariots fell out of use. Fast moving cavalry on the other hand is lighter on supplies, as it is able to forage and raid enemy territory, and it is much faster and flexible than wagon carts pulled by horses, not to mention the aforementioned tactical versatility of mounted infantry in general beyond the issue of strategic mobilization.

However not to leave you totally disenchanted with the idea of the offensive use of wagons there is an idea not quite dissimilar to but not quite the same as yours that did see extensive use in the early modern period, and it was the wagon fort. It was a battlefield tactic that was especially prevalent in Eastern Europe and Central Asia, popularized by the Hussite rebels of Bohemia and later the Ottomans, and it consisted on arranging several wagons, chained together to stop the enemy cavalry from pouring in, and interspersed with artillery and infantry. It was a sort of mobile fortress that could be carried along with the supply train and arranged as the enemy approached, with soldiers shooting from behind and inside the wagons. Ironically this disposition often required cavalry support in order to cover the flanks, screen ahead of the formation and lure the opponent towards it, so even then wagons were still not a good substitute for horses