Disclaimer: I am funded by the NSF.
First of all, having people who probably have little-to-no scientific training, let alone any training or expertise in the field that the grant is in, decide whether a particular research project is "a waste of time" is beyond stupidity. It is equivalent to allowing the average american to micro-manage troop movements on the war-zone, allocate ad hoc rations/supplies to each region of the world, etc. In other words, the solutions provided by the "people" will be far from efficient, far from optimal, and indeed probably just dead *wrong* (i.e. soldiers would starve because people thought "nah they don't need this cooking fuel there, they can use firewood!" or something along equally stupid lines.
Another example would be letting people decide civil engineering matters. Like, let's use the cheap steel for the bridge, it's good enough! Or, let's route this highway right through over there, look it's wide open! -- without understanding all the effects and repercussions that taking any of these actions would have (which a properly trained civil engineer etc. would be more likely to recognize).
Of course, one of the explicit stated purposes of the NSF is to broaden appreciation and understanding of science. It's so important that it's almost impossible to get a grant without being able to convincingly show that your project will have broader impacts outside of your subfield. Of course, the people who would be going through the grants by this YouCut thing wouldn't understand why certain seemingly retarded research projects are important (e.g. why bother measuring the weight of the Earth's core, who cares? I want a new car and a TV.) when really it could be a very serious question that many other projects hinge upon (e.g. geothermal energy, satellites that might be affected by the magnetic field that is generated by the core, etc...). Unless people understand this they would vote against it as wasteful. A lot of projects, and the goal of the NSF, is to make it easy for people to understand these relationships and to respect the science, but I have a feeling that people won't go out of their way to even bother to try to understand it.