One of the big problems that China had from Steppe tribes was their powerful bows that could shoot long distances and their mobility from shooting from horseback. Couldn't various Chinese Kingdoms use their resources to make their own composite bows and train their own horse archers?
edit. At the very least, conventional archers on foot with powerful composite bows should have been completely possible and affordable. They should even be able to fire more powerful bows than what could be fired from horseback, due to being able to use a longer bow.
So I'm not entirely certain which period you're referring to, so full disclaimer: What I'm talking about applies mainly to the Qing Dynasty (1644-1911), who in fact did promote archery, at least on foot.
It is perhaps unsurprising that the Qing should do so, as they themselves were from a horse archery-practicing group, the Manchus (formerly the Jurchens). On the western border, the Dzungars, the last nomadic successors to the Mongols, were a consistent threat until the early Yongzheng reign (1722-36), and so Banner garrisons in the western regions tended to maintain good standards of training and equipment. Archery remained a significant enough skill that military examinations until at least the 1850s consisted primarily of an archery exam and a boilerplate essay – attempted reforms under the Kangxi Emperor (r. 1661-1722) notwithstanding.
However, despite official promotion, archery as a skill gradually declined. An obvious reason would include the defeat of the Dzungar Khanate marking an end to significant offensive expansion (aside from the disastrous Burmese expeditions under the Qianlong emperor). However, the military exams were also declining in both significance and in harshness. Manchus in particular had always been deliberately privileged when it came to advancement through exams, but Han Chinese benefitted as well from the changes that occurred. Promotion through the ranks without examinations became increasingly common, as did purchased posts. Later on, the crisis caused by the Taiping War (1851-64) and concurrent rebellions like the Nian (1851-68), Panthay (1856-73) and Dungan (1862-78) meant that civil mandarins or even rural gentry without government postings were increasingly assigned military commands. As a result, the requirement of archery as an officer's skill became increasingly unenforced.
Additionally, the matchlock musket, despite a couple of drawbacks, was nonetheless a far more cost-effective piece of kit, requiring much less training, and had been in use in China since the 1560s during the Ming Dynasty. Enough would be produced under the Qing that by the time of the formation of the Xiang/Hunan Army in 1853 during the Taiping War, the official distribution of weapons in militia armies was 25% small-calibre muskets and 25% large-calibre jingals. Soldiers could make good use of gunpowder weapons relatively quickly compared to bows, which was especially helpful given the relatively small size of Qing standing army and consequent reliance on more impromptu forces.
You see, although the Qing boasted a nominal 850,000 regular troops, only the 150,000 Bannermen were a concentrated standing force, and even then many of the garrisons, particularly on the coast, began shirking their military duties . The other 600,000, the Green Standard, were required to perform law enforcement duties, operate as couriers and assist in official work, meaning that a province could at most spare 25% of its Green Standard forces for redeployment elsewhere (and this was an exceptional figure attained from a single province at the time of the First Opium War). Moreover, they were also heavily decentralised to prevent too many Han Chinese soldiers falling under the jurisdiction of a non-Manchu official, serving to further depress training levels. In wartime numbers would be bolstered by yong (lit. 'braves'), essentially militias brought on as mercenaries, who therefore likely had no training at all in archery. By the 1860s, consolidated yong units had supplanted both the Green Standard and Banners as the core of the Qing army and begun to adopt Western arms, and the transition to rifles would be finished no later than 1894, spelling the end for wartime archery use in Chinese armies.
In essence, the decline of archery in the Qing army was due to a multitude of reasons, but the main issues were:
Apologies if I haven't quite answered the question you were asking, but this was the best I could come up with given the limits of my own reading.
Sources:
My main source for this was Chapter 1 of Mao Haijian's 《天朝的崩潰》Collapse of the Heavenly Dynasty (1996), available in English as The Qing Empire and the Opium War: Collapse of the Heavenly Dynasty, trans. Joseph Lawson (ed.) (2nd ed. 2016). Others include:
Yes they could, and they did. There was a shift from focus on infantry crossbowmen to horse archery due to steppe tribes (particularly the Xiongnu) during the Han dynasty (206 BC–220 AD). This focus on horse archery with composite bows remained through to the end of Imperial China with the fall of the Qing dynasty (1644-1911). Crossbows would never regain preference again. This was to the extent that by the Ming dynasty (1368–1644), the designs for bronze trigger mechanisms for crossbows widespread in the Warring States period (475–221 BC), had been lost. Even Ming contemporaries considered their crossbows lacking and inferior. In the following Qing dynasty, crossbows were pretty much entirely absent in the military.
However, China had already been producing their own composite bows even before this shift, and it was already the standard form of bow. Many Scythian bows of composite construction have been found both within and outside the historical borders (but within modern day Chinese borders) of the ethnic Han Chinese territories at the time. The Scythians were one of the first horse riders and archers. It is likely that cross cultural exchange in the region took place. In the Zhou dynasty (1046-256 BCE), ritual archery with composite bows were popular. Confucius himself was an archery teacher and he advocated the practice of ritual archery for both military training purposes, as well as to build character, and conduct. Although, crossbows were favoured in the Warring States period and in the Qin (221-206 BC), note that Chinese crossbows of this time were often literally just composite bows mounted sideways on a stock (also note, most, if not all the archers of the Terracotta Army were crossbowmen). As stated before, war with the Xiongnu during the Han dynasty resulted in a shift of tactics, with more focus on mounted archery.
This continued throughout the following dynasties as war with steppe peoples became more and more commonplace. In the Tang dynasty (618–907), horse archery was already an official part of the military Imperial examinations, performed with composite bows at a standard of about 90# draw weight. On foot, 160# was typical. During the Song dynasty (960–1279), came the height of war with the steppe peoples in China. This was reflected in the military - foot and mounted archery were the primary subjects of focus in the military examinations by this time. Composite bows of 160# on foot and 120# on horseback were considered a desirable standard. 145# on foot and 105# on horseback was considered of "bottom rank". Ming dynasty archery remained relatively unchanged, with continuing focus on horse archery to fend off the Mongols expelled from power after the Yuan dynasty (1271–1368). A new reason for this continuing focus was because of the increasing Wokou, or invading Japanese pirates.
Come the Qing dynasty, the Manchus essentially displaced Chinese archery with their Manchu archery, making (mostly) standard their form and equipment across China. Foot and horse archery in the military examinations continued. The Manchus were proud semi-nomadic horse archers and stubbornly stuck to archery despite its increasing obsolescence into the modern era. What gunnery that did exist in the late Qing dynasty was outdated and not widespread enough (smoothbore muskets were still used while rifles and revolvers were already standard in Europe and America). Japan in the early 1600s had already relegated archery primarily to the realm of ceremony. In contrast, it was only in 1901, did the Qing Emperor Guang Xu state that the military examinations as a whole (including archery) were "useless" and of "malpractice", and proclaimed that they be permanently abolished.
My main source for the history of archery in China is from Chinese Archery by Stephen Selby. The source goes through a collection of primary sources, many of them archery manuscripts. For further reading, do check out 射義, 射道, ATARN, fe doro - Manchu Archery, and DEBTELIN.