Tribal nomadic societies can sustain a greater percentage (around 20% which is basically all military age males) of the population as warriors. Once you have sedentary agricultural population, money based economy and specialisation the number if full time soldiers supported becomes a much smaller number (maybe 1%). Once you have cities (or other centres) supporting specialist classes you have to produce a surplus from the agricultural sector and transport it to support the non agricultural population. Sure a nomad society can mobilise a greater percentage of the population but it's structure cannot support a large number of specialists. Literate society, metallurgy, engineering, medicine, science. Sure the mongols adopted and used specialists in siege engineering, but this was a parasitic relationship as the nomad culture of the mongols was incapable of developing such things. Sure they acquired/trade metal armour and products but they did not develop such as an internal capacity.
The main point is that number of warriors in the main question and relives and society structure, the 'flat' structure of tribal society contrasted with the specialisation of cities/agriculture/monetary economy. The number of archers is another question, which you can look at the relative importance of archery in different situations.These factors dominate the question rather than methods of bow manufacture or warrior skills.
I question the Idea that Europeans soldiers were only trained for one form of combat, full time professionals had a range of skills, long bowmen proficient at hand to hand, Knights fighting on foot. The Mongols had a small range of types of warrior fielded and there was a high degree of interchange in the skills. The European armies existed in a more complex world of a much higher range of skills, were total competence across all of them was pretty unlikely to be achieved by individuals.
European armies in the main in this period were full time professionals of which there was generally an over supply. Recruiting more was generally just a matter of money which tended to run out.
I'd question exactly the interchange of light and heavy horsemen in mongol armies. Mongol heavy cavalry had horse armour and heavy per coal armour, was there specialisation of horses rather than riders? Have we got concrete documentation of heavy and light mongol warriors interchanging roles? While a lot of the skills are interchangeable they may well have been some status involved that in societal terms it simply was not done.
Bottom line European armies fielded less archers because archers were not as important in European warfare at the time. Archers typically are a large factor in warfare (pre gunpowder) when there are either large amounts of cavalry or unarmoured infantry present. In the absence of this, archery is just less important.