So we are talking about differences in scale, not kind.
Saying “We can’t build that - it would need to be 4 to 16 times bigger than anything we have built!” is very different from saying “we can’t build that - we don’t fully understand the underlying principles yet, and it is possible we will never be able to build one”, like we might about a fusion reactor, or “we can’t build that - as far as we know the laws of physics do not allow for it to exist”, like we might say about a warp drive.
Not to mention, this is one thing that needs to be 100% foolproof. One crashed rocket of nuclear waste would be fairly bad news for wherever it landed. Far too many things can go wrong; just from the US space shuttle project, 2 failures out of about 135 missions (you could argue just 1 failure, since one was on re-entry which would not be a concern in this use case).
So, a 1-ish percent failure rate. Not sure I’d be comfortable with that.
Best to just send it up the space elevator 
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What happens if the space elevator breaks? 
Absolutely. And the counterweight could be made adjustable, so that the centre of mass could be shifted to accommodate different amounts of traffic up and down the elevator. You could also use the far end of the elevator as a sling, to propel payloads around the Solar System without using propellant.
If the elevator could be built it would be very useful, but the difficulties seem insuperable.
(Or, to bring things full circle, a space elevator)
I think you misunderstood; my observation that “…scaling up solar panels and the associated inert mass by a factor of 4 to 16 to get the same power as to intercept a Near Earth asteroid” was just referring to the comparison in the scale of the solar power system operating at the orbit of the asteroid belt to one at ~1 AU, which is significant when talking about having to scale up required power by orders of magnitude, especially with a large distributed structure like a solar panel array. But the change in the mass to be returned—from a small boulder-sized chunk for ARM to a whole asteroid of sufficient mass to provide a counterweight, e.g. tens or hundreds of thousands of tons—is many orders of magnitude more difficult just in terms of propellant usage and impulse, much less the practicalities of how you would apply thrust to such a large and very likely highly friable aggregation, or how you would control thrust with unexpected changes in the mass properties as the payload flows and shifts under even small accelerations.
At a certain point, “differences in scale” are differences in kind. A system capable of redirecting even a moderate sized asteroid would require both an extensive amount of in-space assembly as well as new techniques and far more automation than our experience with the biggest in space structure (the International Space Station), notwithstanding a propulsion system that could operate on that scale. These are easy things demonstrate conceptually in a slide deck or an animation but far more difficult to realize in practice.
Stranger
To the lump of nickel the years are as nothing. To the investors paying for all this, the years are as 10% lost ROI.
You aren’t accounting for the thriving market for asteroid futures. The lump of nickel may change hands dozens of times while still en route!
It is even worse than this; bringing a large quantity of precious and industrial metals back to Earth would saturate the existing commodities market, so in order for such a scheme to be remotely viable the materials would have to be doled out gradually into the market akin to how the De Beers Group created an artificial scarcity for gem-quality diamonds, which means the investment is amortized over a long period rather than getting a quick return. Even worse, if there are competing enterprises they would either have to form a cartel or would drastically undermine each other.
Either way, in order for such an enterprise to be remotely viable, the technology and infrastructure would have to be available to perform the extraction and processing in space, which is undoubtedly beyond the capitalization means of any commercial interest operating on a timeframe of a few years; this would be an investment requiring decades to provide a net return (if ever) for terrestrial usage, by which time the market demand for that material may have dramatically changed. The only real investment that makes sense is to use these materials for development in space.
I once conceived of and did research for a story involving the competitive economics of space mining in the asteroid belt based upon the demand for and celestial mechanics of delivering various commodities to solar orbiting colonies using only low efficiency solar electric propulsion. I got as far as building a predictive model for “potentials” before discovering how incredibly chaotic such a system would be given the availability of many material resources and the complexities of being competitive with constantly changing ‘trade routes’. Basically, any resource could end up being completely worthless by the time you got it somewhere that it would theoretically be used due to market saturation. So hypothetical “space economics” is far more complicated than you might expect.
Stranger
I think it would be quite a while before asteroid-mining (for use on Earth, anyway) would be cheaper than just strip-mining right here. For use in space colonies, it’s a different scenario - no gravity well to escape if shipping the materials up from the planet, for example - and you’ve presumably (hopefully!!) solved the problem of reliable life support and maneuverability in vacuum.