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.

If you have additional information beyond the first derivatives then by all means use it. Use all the information you have. However, in general you need more information to get an equally good approximation to higher order derivatives. Cross terms especially seem like they would be very difficulty to gauge empirically. In light of that I would be very skeptical of high confidence estimates for higher order terms, especially if they conveniently twist the math to allow for a desirable outcome.