I think when you provide people with absolutely no context like this, you're bound to get stupid answers. I feel like if you instead gave a list of things the government should do and asked people to allocate money to each, you would at least get answers that are sane, if not accurate.
And honestly, knowing this kind of stuff is kind of irrelevant anyway. The whole reason we have a representative democracy is so that ordinary citizens can delegate the specifics of government to experts.
I think it's more of a matter of disinformation; if US citizens are radically disinformed, that's another barrier to participating in allocating our budgets. The system obviously favors decisionmaking by wealthy elites.
(Affording your own decent healthcare is enough of a feat in the US, that if you're wealthy enough to chip in for some of other people's healthcare too, people think you must be Batman or something.)
As for Bill Gates in particular, apparently one problem is his support for intellectual property, though his foundation. Many consider IP (particularly patents) a major problem for world health; and certainly the US knew better than to respect other countries' intellectual property, while it was developing. (To Charles Dickens' consternation.) (http://keionline.org/microsoft-timeline)
And honestly, knowing this kind of stuff is kind of irrelevant anyway. The whole reason we have a representative democracy is so that ordinary citizens can delegate the specifics of government to experts.