I think this question is founded on an unrealistic appraisal of the utility of horse archers, as well as a questionable grasp on the timeframe at work here. I do appreciate you're not insensible to the issues of society and military mobilization, though; a large host of men serving as horse archers require a vast social support system, after all, and this cannot be changed on a whim. That said, there were many other factors that made horse archers less attractive than they may seem, even when modern states had the financial muscle to train and equip men to fight the way they wanted. I'll be focusing on the period of the 16th-18th centuries, just for future reference.

Particularly in the era of firearms and standing armies, the horse archer was of very limited use. I don't know what you consider 'the advent of reliable gunpowder weaponry', but by the beginning of the 16th century, the bow was obsolete in Europe; the superior accuracy, range, and killing power of firearms relegated it to local defense by militias at best. Its chief virtue was that it was a remarkably cheap weapon, no more expensive than the arrows it shot, but troops armed with bows were at the greatest disadvantage against firearms. The deficiencies in bows were magnified even more by the special challenges of shooting from horseback compared to foot archers, making it even more difficult for such a force to gain missile superiority.

Your example of Gustavus Adolphus is particularly odd, as his tactical system relied heavily on the most modern firearms, including mobile artillery and linear formations firing massed salvos of musketry. This was the way of the future; his deployment at Breitenfeld would be easily recognizable to commanders up to the end of the 18th century. Chastising him for ignoring 'the most effective pre-gunpowder unit' would be like chastising the US Army today for neglecting to include mounted knights in its order of battle. That said, his cavalry force was well equipped with missile weapons; the typical horseman in his army carried two pistols for use in battle and a carbine for dismounted fire.

By his time, firearms were certainly ascendant, to the point that many societies which had previously relied heavily on mounted archery had come to incorporate gunpowder weapons into their arsenals on a grand scale. While the Manchu, the Japanese, the Chinese, the Mughals, and the Ottomans all retained the horseman's bow to a greater or lesser degree, firearms increasingly formed the core of their armies, and their great conquests relied on them to the point that many historians label them 'gunpowder empires.'

Besides the deficiencies of bows compared to firearms, employing a horse archer army in the style of the steppe empires carried problems of its own. The most obvious is that Europe is not a steppe, and a good general could easily find terrain in a theatre of war wholly unsuited to the great wheeling evolutions of cavalry on the plains. Even if he couldn't, armies by this time were well practiced in the art of castrametation; a fortified camp on well chosen ground could be considered invulnerable to attack. When mounted armies tried to assault wagon forts in eastern Europe, they were often repulsed with heavy losses. However, there several further linked logistical issues with the employment of horse archer armies in the European context.

An army of horse archers would be much harder to supply in action; in addition to all the men, they could only campaign in regions where they can gather large quantities of forage, which limits the energy and flexibility of warfare. A horse's feed for a day weighs ten times a man's, and a horse will die from privation far sooner. Since most European warfare was fought for the possession of territory, sieges were the order of the day; only control of fortresses would allow a state to impose its will on a region. Supplying an entirely mounted army in one place for an extended period is even harder than for one of foot; unable to move, a mounted army would quickly exhaust the available local supplies, and supplying them by wagon train would be an absurdity with the sheer weight of fodder.

Horses are more expensive to maintain than men, so with the financial limitations of Early Modern states, funding such an army would necessitate a much smaller force than they could otherwise field. An army of 15,000 horse archers would be much weaker than one of 40,000 infantry, 6,000 cavalry, and 80 artillery pieces, these two hypothetical forces costing roughly the same by the dawn of the 19th century in Clausewitz's estimation is to be believed. Moreover, the second force would require less weight of provisions each day.

Lastly, we have an intriguing example of what did happen when European armies fought bona fide horse archers from the Eurasian steppes; they were Baskirs levied by the Russian tsar to fight against Napoleon, and lacked so much as a sword to accompany their bow. General Baron Marcellin Marbot was a cavalry officer serving in the French army during the Napoleonic Wars, and towards the end fought against them at the battles of Dresden and Leipzig. I'll let his assessment speak for itself.

Their efforts were chiefly directed against Sébastiani’s cavalry, and in a moment the barbarians surrounded our squadrons with loud shouts, letting off thousands of arrows. The loss these caused was slight, for the Bashkirs are totally undrilled and have no more notion of any formation than a flock of sheep. Thus they cannot shoot horizontally in front of them without hitting their own comrades, and are obliged to fire their arrows parabolically into the air, with more or less elevation according to the distance at which they judge the enemy to be. As this method does not allow of accurate aiming, nine-tenths of the arrows are lost, while the few that hit are pretty well spent, and only fall with the force of their own weight, which is inconsiderable; so that the wounds they cause are usually trifling. As they have no other weapons, they are certainly the least dangerous troops in the world.

I hope I've illustrated some of the major problems with the idea of horse archers in Early Modern armies. They were neither particularly combat effective nor practical in overall terms.

I've based this assessment on a pretty broad study of European warfare during the Early Modern period, so I'd recommend starting with some of these works to get a general feel for the period.

MS Anderson, War and Society in Europe of the Old Regime

Michael Axworthy, The Sword of Persia

Jeremy Black, European Warfare: 1660-1815

David Chandler, The Art of Warfare in the Age of Marlborough

Christopher Duffy, Military Experience of the Age of Reason

Henrik Lunde, A Warrior Dynasty: The Rise and Fall of Sweden as a Military Superpower, 1611-1721

John Lynn, The French Wars: 1667-1715

Geoffrey Parker, The Army of Flanders and the Spanish Road

---- The Military Revolution: Military Innovation and the Rise of the West, 1500-1800

Henry Jameson Webb, Elizabethan Military Science

Hey there,

Just to let you know, your question is fine, and we're letting it stand. However, you should be aware that questions framed as 'Why didn't X do Y' relatively often don't get an answer that meets our standards (in our experience as moderators). There are a few reasons for this. Firstly, it often can be difficult to prove the counterfactual: historians know much more about what happened than what might have happened. Secondly, 'why didn't X do Y' questions are sometimes phrased in an ahistorical way. It's worth remembering that people in the past couldn't see into the future, and they generally didn't have all the information we now have about their situations; things that look obvious now didn't necessarily look that way at the time.

If you end up not getting a response after a day or two, consider asking a new question focusing instead on why what happened did happen (rather than why what didn't happen didn't happen) - this kind of question is more likely to get a response in our experience. Hope this helps!