What to Cut and Why
As a result of YouCut (based on the ordinary citizens selecting government programs for termination), Scott Aaronson has sarcastically asked which scientific fields count as “hard” and therefore worthy of support. I will supply a non-sarcastic response.
One reason for academics to try to do without external support as much as possible is that such support produces the suspicion that they’re trimming their results to the agenda of the supporters. For example, it's common for critics of the Anthropogenic Global Warming hypothesis to claim that the supporters are merely looking for grants. (This type of theory is not limited to one side of the political spectrum.)
To the extent this happens, I don't think this is a matter of researchers asking supporters what answer they want and then producing it. It's more likely a matter of asking for a grant to find out if X is true (where X is something the supporter wants to be true) and then not publishing if the research doesn't produce a positive result.
One possible way to fight that is to preferentially support fields where negative results are mentioned more often. That might at least produce some peer pressure to report those negative results. It might even cut down on conspiracy theories.
In the specific case of research supported by corporations, anything that mentions “rent seeking,” should be considered a negative result.