It's not at all accurate to characterize medieval infantry as "Levied Peasants;" by and large, soldiering in medieval Europe was the exclusive purview of people who owned arms and armor and knew how to use them. These two requirements typically meant that military recruits were landowners. And in the specific case of archers (although I'm certainly much more well-versed in the peculiarities of Italian history) I'm fairly confident that across Europe archers were also drawn from freemen, typically small landowners, from regions where people already had strong traditions of archery. In some places, like England, these traditions were widespread; in other places, like Italy, they only really appear in areas unsuitable for intensive agriculture, like mountain valleys and marshlands (medieval Venetians chronicles, for this reason, make frequent mentions of archery paraphernalia).

In fact, it's interesting to point out how late medieval Italian political organizations were fairly unique in the way they managed to involve multiple segments of society in their armed hosts, and within this approach the disproportionate presence of urban recruits. Although multiple european states also involved urbanites in their armed forces; with places like Portugal and Burgundy standing out (the latter was particularly interested in involving burghers proficient in the use of crossbows) these were compliments to a broader force of recruits consisting of property owners. In short, no other states were as reliant on urban recruitment as those in Italy. This is, of course, a consequence of the fragmented urban-centric political organization in Italy: when an urban council like that in Florence or Milan decided to go to war, the citizens who were most represented in the council would be expected to be the first to mobilize and bear arms. For these purposes, "Citizens" generally means "Upper-Middle Class Males;" but that would include most members of the more prosperous guilds, the wealthiest among the members of minor guilds, and suburban landowners. On top of this, even though the poorer members of society could not afford to arm themselves autonomously, organizations based around neighborhood parishes or trade guilds would try to pool resources in order to contribute fighting men. In the largest cities like Venice or Milan there were even more elaborate systems in place to draw recruits from the poorer members of society: in Venice, lotteries determined who was drafted into the navy (and every draftee was drawn with three neighbors who would pay for his arms and armor) while in Milan great lords called Capitani who composed the polity's executive council were expected to provide arms and armor for those who could not afford them in their neighborhood. That being said, even with these systems in place the difficulty of mobilizing substantial numbers of people from the minor guilds and/or lower middle class meant that there was typically a disproportionate amount of upper and middle class citizens present when Italian hosts set out in force. Plus, this system generally made for poor-quality fighting men: Italian armed hosts performed poorly when pitted in the open field against French or German forces whose recruits were drawn from a more traditional and typical warrior-aristocracy.

It is also worth noting though that in Italy, as in the rest of Europe, the wealthiest and most powerful would normally employ retinues of semi-professionals who do not fill the mold of the property-owning fighting man: "Sarjeants" were servants who were employed by great lords and trained in the use of arms, while great magnates could be expected to retain a varying number of landless knights in and around their household. Sarjeants were particularly useful when Italian polities mobilized in force, keeping order amongst the line of battle would largely be their responsibility of they were numerous enough. However martial training and drilling would generally be in the purview of community leaders, typically members of the guilds or parishes who had more experience in warfare. In other european countries, Sarjeants coupled with less wealthy freemen (or simply less skilled freemen) would instead compose the bulk of the infantry.

It's also true that the composition of a single armed force in medieval Europe was largely need-based. Although it's true that cavalry was typically composed of fighting men from wealthy backgrounds with the resources to afford horses and equipment, if the situation required it they would dismount (as an example, the German Emperor Frederick Barbarossa was defeated at Legnano by an Italian host whose cavalry had all dismounted).