I think that one of the major problems isn’t even the zombies as such, but the collapse of infrastructure that we rely on to maintain our current populations. Transportation and associated trade. Electrical power. Fuel. Fertilizer.

Like, even if you omitted zombies from the equation and took those away, the population that could be supported is probably a whole lot lower than the present population. Pre-industrial societies were not very populous compared to current ones.

https://en.wikipedia.org/wiki/Carrying_capacity

The carrying capacity of an ecosystem is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as the environment’s maximal load,[clarification needed] which in population ecology corresponds to the population equilibrium, when the number of deaths in a population equals the number of births (as well as immigration and emigration)

Human carrying capacity is a function of how people live and the technology at their disposal. The two great economic revolutions that marked human history up to 1900—the agricultural and industrial revolutions—greatly increased the Earth’s human carrying capacity, allowing human population to grow from 5 to 10 million people in 10,000 BCE to 1.5 billion in 1900.[47] The immense technological improvements of the past 100 years—in applied chemistry, physics, computing, genetic engineering, and more—have further increased Earth’s human carrying capacity, at least in the short term. Without the Haber-Bosch process for fixing nitrogen, modern agriculture could not support 8 billion people.[48] Without the Green Revolution of the 1950s and 60s, famine might have culled large numbers of people in poorer countries during the last three decades of the twentieth century.[49]