You also have to believe that population growth is fully exponential. But in practice it looks more like a logistic curve, and the factors that slow it down are a lot less dramatic than resource starvation or violence.
Additionally, it is reasonable in the 1790s to assume exponential population growth. It would not be until the 1830s that Verhulst would even model it as a logistic curve. Moreover, human population growth really was on a roughly exponential trajectory until very recently.
Obviously we did not experience such catastrophe. But it's also a bit wrong to call it "invalid" when it was a completely logical conclusion to make. If we're going to allow him to invent a magical technology that addresses it in 1798, then all reason goes out the window.