I have no useful information here, so a uniform prior seems reasonable, in which case the analysis from a linear model holds.

I'm not sure what the uniform prior means in this case and how the conclusion follows - can you expand?

But anyway, granting this for the moment, in an actual real-life situation when you contemplate actual charities, you do have all sorts of useful information about them, for example the information that allows you to estimate their effectiveness. This information will probably also throw some light on how the effectiveness changes over time, and so let you determine whether the linear approximation is good.

This seems especially true when the differences in first derivatives between different charities is large, such that a second order correction would have to also be very large in order to sway the analysis.

I agree that when first derivatives are wildly different according to your utility function, it's a no-brainer (barring situations with huge second order effects that'll show up as very weird features of the landscape) to put all your budget into one of them. What I object to is slam-dunk arguing along the lines of "Landsburg has a solid math proof that the rational thing to do is to take first derivatives, compare them, and act on the result. If you don't agree, you're an obscurantist or you fail to grok the math".