My thinking is that donating during drawdowns might be particularly bad
This is true, and the standard deviation fully captures the extent to which drawdowns are bad (assuming isoelastic utility and log-normal returns). Increasing the standard deviation is bad because doing so increases the probability of both very good and very bad outcomes, and bad outcomes are more bad than good outcomes are good.
Is it actually the Sharpe ratio that should be maximized with isoelastic utility (assuming log-normal returns, was it?)?
Yes, if you also assume that you can freely use leverage. The portfolio with the maximum Sharpe ratio allows for the highest expected return at a given standard deviation, or the lowest standard deviation at a given expected return.
This is true, and the standard deviation fully captures the extent to which drawdowns are bad (assuming isoelastic utility and log-normal returns). Increasing the standard deviation is bad because doing so increases the probability of both very good and very bad outcomes, and bad outcomes are more bad than good outcomes are good.
Yes, if you also assume that you can freely use leverage. The portfolio with the maximum Sharpe ratio allows for the highest expected return at a given standard deviation, or the lowest standard deviation at a given expected return.