It can't
directly distort science because a) people are applying for grants to conduct research.
They don't yet know what the results are going to be, unless you accuse the scientific community of massive fraud. And, b) because research grants, at least the ones I know of which are given out by NSF and NASA, are not awarded based on what the government thinks but on how the proposal is rated by an independent panel of peer-reviewing scientists.
You can of course imagine second-order effects whereby the selection of the members of the peer-review committee is skewed by the program officers, picking panels that they think will give money to the programs they want. But if they consistently picked committee members that are out of the scientific mainstream, I think you'd hear a howl from the scientific community. I've heard plenty of complaints about the awards from people who weren't funded, but that the panels were stacked by people with predetermined ideas of what results should be obtained is not one of them.
As to your K-12 example, do you know for a fact that there are (competent) researchers who have applied for grants to conduct that study and were rejected? If not, you have no case.
If you think the government orders what research be done, you are woefully uninformed of how the vast majority of research is conducted in the US. Aside from a small number of researchers working in government facilities, research grants are awarded based on proposals from independent researchers. If no one applies to conduct a study, it won't happen no matter how much "the government" wants it to. And as far as I know, there is no government-run research facilities in the field of education.
And finally, in case you haven't noticed, Al Gore has not been affiliated with the government for a decade...