If you instead kept that $100 to spend on yourself, you would give the government $40 and you could spend $60.

In the case you give it to charity, the development/executive staff pays income taxes on that $90, so if their marginal rate is the same as yours (assuming they're highly paid), they pay $36 in taxes, have $54 to spend, and $10 goes to programming. If it goes to staff paid under $147K, they pay their (lower) marginal rate on it, but the full 15.3% amount for Social Security and Medicare is paid on those funds.

It seems like the government is only out ~$4 ($40 less from you; $36 more from someone else), with ~$10 available for programs.

(The largest loss is that you lost $60 of spending power in exchange for only $10 in programming while development staff got an extra $54 of spending power. You lose way more than the government loses.)