Since the OP focuses on Gates’ wealth, I guess we’re supposing some kind of “Day of the Jackal” scenario where Gates hires a super-assassin to do the job. I see at least two problems.
The first is locating such a person. I don’t doubt that he could do it, but I think it would be difficult for him to do it without attracting unwanted attention, as others have noted. And in this case, “unwanted attention” could have very harsh consequences, even for a guy who’s extremely wealthy.
The second problem, I would think, is convincing such a person to take the job. A hired assassin who tries to kill the President (successfully or unsuccessfully) will literally have an army of people after him, with unlimited resources and very few restraints on their behavior. I suppose it would be possible to pull off the job so cleanly that the authorities are simply baffled about who did it, but that seems like a long shot. And if they know who you are, your only shot at survival is to find some really desolate place and adopt Osama bin Laden’s lifestyle for the rest of your days. A religious fanatic might put up with that, but an assassin for hire would presumably not be interested. What good is a billion dollars if you can’t ever spend it?
Actually, getting paid might be another problem. Any kind of electronic payment will most likely get traced and taken away. (I don’t care how confidential banking is supposed to be in the Seychelles or the Isle of Man; when Delta Force crashes through the door, your banker’s going to give it up.) And good luck carting around 5000 cubic feet of cash while you’re hiding in the Gobi desert.