Can you explain why the naive assumption is wrong? Because I actually can't, not without more thought and research than I've given it.
Gates can give substantially more money if he leverages it to grow over time. I would wager that it is going to take decades if ever that the Gates fund will hit $0. Yeah it's not as flashy as "Died with $0 of his fortune left", but under the hood the net good done is orders of magnitude greater.
If you can do that in perpetuity, then it actually means you're giving away none of your wealth away. You're selling the time value of money back to the people that generated it for you.
You've essentially taken out a giant loan, but the interest is being borne by the people who generated that wealth for you in the first place. And your "charity" amounts to sometimes reimbursing those people for paying the interest.
> Then your net worth is $110 and when you give away $10 your net worth goes down $10.
Gates' wealth fluctuates all the time. There was a time when he was rather low due to his donations - multiple 10s of billions down. He got back over $100B from cca $50B - that's $50B more for charity than if he just donated everything he had left.
For instance, Chuck Feeney gave away billions until it he was left with only $2 million (or so it was reported).
If they truly wanted to just give it away, Chuck Feeney offer a straight forward model for doing so. You just do it. If "giving it away" merely means putting it into a foundation you control, many people would feel that little has, in fact, been given away.
And the PR? Let the guy have his fame, he's really worked for it.