Soldiers absolutely had wooden armour. It was quite common on the battlefield as well. It was called a shield. You could hide behind it. Hit the enemy with it. Rest it on the ground when you were tired. Throw it away when it broke. Best of all it fended off arrows and blows well away from your body where they did far less harm. If you had the money to spend on the materials for wooden armour, you had the materials for a shield.

But let's say someone really wanted to try this out. After all there were some cultures throughout history who had armour made of bone, wood and wicker.

We think of steel as being very heavy. However here we can more precisely describe it as dense. It's heavy for it's volume. However, steel is also exceptionally strong, so requiring less volume to produce the same structural strength as other materials. For instance, a decent sword at the time weighed around 3lbs. A thin blade, usually fairly well balanced and wouldn't break easily under the stress of hitting things with it. A chunk of oak around 3lbs is about the size of an axe handle before I'd go about hitting things with it as hard as I could. We're talking the difference between 1/8" of steel and 1" of wood to do the same thing, and steel does it better.

Plate armour by the end of it's reign was sitting around 1/16" to stop a sword blow. Was well constructed to conform to the body as to not hinder the movement of the wearer so as to make him an effective fighter. If we replace that with some oak boards of a similar weight we're adding an extra half inch of bulk to our peasant soldier, on top of the gambeson. This homemade contraption probably wasn't very well put together, is full of holes, is hanging off your body awkwardly and just gets in the way of your ability to fight. Something you likely weren't the best at to begin with. Being a peasant and all. It still hurts like crap to get hit while wearing it. To top it all off, it's still not nearly as good as steel or iron of the same weight.

No no no. What you wanted was a shield. A nice stout piece of cover to stay behind when the various pointy bits of the enemy came crashing into your lines. Much more practical for the part time levy.

Your question is predicated on a number of assumptions that you present in a very certain way. Where are you drawing these from? I really think you underestimate the number of well-equipped warriors to half-armed peasants.

As has already been pointed out, I think you underestimate the extent to which warfare was a business mainly in the hands of people who owned land and whose resources and time allowed them to buy arms and practice with them. Sometimes fighting men were recruited with the promise of monetary reward, other times social and political ties were created by the granting of some sort of privilege that came with the duty to fight. Systems also appeared where men could be offered room and board in the household of a great lord, or have been granted their own plots of land (sometimes with renters or bondsmen to work it). What is consistent is that soldiering in Medieval Europe was the purview of people who had the ability to own arms, armor, and the free time to dedicate to learning how to use them. In my own area of expertise, Medieval Italy, warfare was initially delegated to an entrenched landowning aristocracy who administered a second tier of land-owing vavasours.

When, by the thirteenth century, urban professionals had become politically enfranchised and their leaders inserted and and integrated themselves with the nobility, members of the upper middle class like notaries, lawyers, bankers, and moneychangers would also be expected to purchase and train with their own arms and armor. This was particularly visible in communities like Florence and (sort of) Venice, where the government system was heavily skewered in favor of the urban professionals. In other places, notably Lombardy, in spite of urbanization and similar economic development there remained an influential warrior-aristocracy which drew its wealth from land ownership; and it is perhaps because to this too that Lombard cities like Milan and Brescia developed large and flourishing communities of blacksmiths specializing in arms and armor (Via Armorai and Via Spadari are still streets in Milan, although ptredictably their function has since changed). And even the less prosperous and more "normal" urban professionals and artisans would eventually be expected to own arms and armor, with the crossbow a particularly favorite weapon: expensive weapons that required relatively little training, the perfect combination for urbanites with disposable income but, unlike the landowners, little free time on their hands.

That's not to say there were no attempts to expand the socioeconomic pool of military recruits. Organizations based around neighborhood parishes or trade guilds would try to pool resources in order to contribute well-armed and trained 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 pre-13th century 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 combination of warrior-landowners and seasoned professional retainers. It is also worth noting though that in Europe, and Italy too, the wealthiest and most powerful would normally employ retinues of semi-professionals who do not fill the mold of the prosperous property-owning fighting man: "Sarjeants" were servants who were employed by great lords and trained in the use of arms, while great magnates could also be expected to retain a varying number of landless knights in and around their household.