I guess the authors never heard of a cometary orbit. The Sun is indeed at rest relative to the solar system as a whole. Nevertheless--it moves (to coin a phrase). An object traveling through the Solar system (even if it's only interplanetary) has a relative motion that is 'not quite' the same as the sun's. Increase this velocity sufficiently and a gravity assist from the sun becomes possible. To do this efficiently you need gravity assists from multiple planets. However, if you're going to do that, I agree--just use the planetary assists to throw you out of the Solar system.
Is the Ion reaction mass minuscule in absolute terms, or only in comparison to chemical rockets? I need to try computing how much delta-V is needed for a slow spiral into Sol. But the first realistic Ion figures I find indicate that the required reaction mass still far exceeds the mass of the payload and vehicle. To guarantee that the waste never gets back to our neighborhood, we need to make sure the mission doesn't fail or end until its aphelion (the orbit's farthest distance from sun) is well inside Mercury's orbit. That is still a very tall order.
What is special about cometary orbits? Besides that fact that the long period comets are already very close to being ejected from the Solar System, at least compared to Earth's orbit. Getting from Earth to these comet orbits requires almost as much energy as escaping entirely. The main problem with planetary assist is the risk of error or failure with a load of nuclear waste. The political price tag is very high.
Here's a link to the Deep Space 1 FAQ on ION propulsion. The reaction mass for an ION drive would be measured in lbs rather than the tons required for chemical rockets.
As I suspected, the reaction mass is minuscule only in comparison to chemical rockets, not in absolute terms. The link you provided indicates the exhaust velocity is about 100,000 fps (30km/s). This probe had 179 pounds of 'fuel', good for a delta-V of 14,800 fps for the 824 pound craft. (My equations suggest that 19,600 fps was possible, if operated at maximum efficiency. But I understand this type of engine can be run at higher thrust with reduced efficiency.) A direct drop into the sun from Earth requires a delta-V of about 98,000 fps (I do see an error in my earlier quick calculation), requiring 'fuel' (actually reaction mass) of about 1400 pounds for the same size craft. But because this engine produces low thrust for an extended time, that path is simply not possible. The slow spiral route into the sun, as suggested earlier, requires even higher cumulative delta-V, due to one of the cruel jokes played by orbital mechanics. The closer one starts to the sun, in a circular or near-circular orbit, the more fuel it takes to get to it. (Another of the cruel jokes is that it takes far more fuel to get to the sun, even by direct drop, than to completely escape from it.) My preliminary exploration suggests a delta-V of 290,000 fps just to spiral down to Mercury's orbit. That boosts the needed 'fuel' load for a DS1-size craft to more than 14,000 pounds, i.e. multiple tons. This is why all the space probes use planetary gravity assists instead of direct or spiraling paths. Getting it all the way to the Sol's surface, requires, um, more xenon than we can acquire. But I do need to review these estimates before anyone quotes any of these figures to an actual rocket scientist. The nuclear waste we would want to get rid of, along with its launch-accident-proof containment, is orders of magnitude heavier than the DS1 craft. I believe it is simply better to keep the nuclear 'waste' down here, where future generations can re-mine it extract its unused energy.
It would seem to me that any high gravity body will do for a potential dump. Sol is a great idea but Jupiter makes for a fine alternative. Though of course both would be incredibly expensive, maybe refining the waste, and sifting what is usable and not and jettisoning the debris to one of the afforementioned bodies would be a viable solution.