There's also economic problems with a peasant republic. In pre-industrial societies, there are only two significant factors of production, land and labor. The supply of land is pretty much fixed, while the labor supply varies with the peasant population. And for the most part, the better off the peasants, are, the more kids they have. You can get more crop from a given amount of land by working it with more peasants, but past a certain point you face diminishing returns so the crop-yield-per-peasant declines. Until capital arose as a third significant factor of production (allowing labor to become more productive without adding more men or more land) in the 17th and 18th centuries, what you generally saw was a cycle of:
- Good times following a technological shift or population shock when there were relatively few peasants and per-peasant crop yields produced a relatively high standard of living.
- The relatively well-off peasants have more kids and the population climbs to carrying capacity, where the per-peasant crop yield barely suffices to feed the peasant's family and pay the peasant's obligations (taxes, rents, etc).
In your typical medieval society, much of the land was owned by members of a hereditary nobility, who rented the land to their peasants. The rents represented the society's agricultural surplus, out of which nobles payed for governance, infrastructure, the church, the military, etc. The military in particular, since the nobility generally doubles as a military caste because 1) they need to be able to fight to hold their land against invaders, bandits, and squatters, 2) rental income is passive, which allows them to train for combat as a full-time job (a necessity with medieval hand weapons if you want to be good at it), and 3) they had the means to hire skilled craftsmen to make high-quality weapons and armor for them and to train and keep warhorses.
Because the nobles were skimming off rents from the peasant's crops, the subsistence level at which a peasant can no longer keep up with his expenses is higher, so the population of peasants leveled off at a lower level where there was still an overall surplus rather than rising to the absolute Malthusian carrying capacity. Without the nobles (or someone in the nobles' place) skimming off the surplus, a few generations later there would be no surplus.