The spend on fundraising does go towards the mission, because it increases the amount of money for the mission. You're making the incorrect assumption that donations are constant.
Imagine that a charity hiring a fundraiser for $50k garnered $150k of additional donations. Let's say without that fundraiser, they got $100k in donations. So with the fundraiser, they got $100k + $150k - $50k = $200k net to spend on the charitable purpose.
20% of all donations go to the fundraiser's salary while she's employed. But if the fundraiser is sacked, then 100% of donations go to the mission, yet the mission takes a loss of 50%.