Q:

Can we destroy an asteroid by spinning it?

In a YouTube video Colonizing the Solar System, part 2: the Outer Solar System, Isaac Arthur who is a YouTuber and also a physicist says something like "you don't spin an asteroid for artificial gravity because it will cause it to fly away". 
In more than few instances, he also explains how easy it is to dig an asteroid as it is more of a loosely held together dust and debris than something solid. So,

Why doesn't an asteroid turn to dust as soon as it enters the earth's atmosphere?
Can't we spin an asteroid in order to pulverize it?
What about mechanically grinding it, if spinning is not an option? When it is so soft, would that mean we surely can send moderately small spacecraft that can grind few cubic kilometers in few years?

A:

Not really. If an asteroid that was a loose collection of rubble was spinning, it would fall apart. The question is, how do you get it spinning? And would it do any good anyways?
To make it spin one would have to apply an asymmetric thrust. The problem is, most likely if you can do this, then it will actually end up only affecting the area you are thrusting, still leaving the main body of the asteroid intact. 
An easier solution almost certainly would be the nuclear option. With a loose collection of rocks, digging into it should be easy enough, just place the nuclear weapon in the middle of the pile and it would scatter quite effectively. This would be quite a bit easier then trying to spin the rock, and impart more energy. The amount of energy required to blow something up is known as the gravitational binding energy. Suffieceth to say, a 100 MTon bomb could blow up a 10 km asteroid, barely. That is the largest nuclear weapon ever built. 
And the more interesting question, would it do any good? That is difficult to tell really, but the answer is probably not. A single dense asteroid will hit the Earth mostly intact, which would cause local severe damage, but probably not global damage, unless the rock was huge. A loose collection of objects hitting the Earth all over, or even worse dust, could cause a different problem. If all of that dust hit the atmosphere, it would probably stay up there. It is very difficult to get things from the upper atmosphere down, thus any dust reaching that high could stay up there for years. This could cause a massive cooling event, similar to a volcanic winter. That could have global repercussions. Note that it would take a large amount of dust to cause this kind of problem, 60 tons of dust enters Earth's atmosphere every day. If, say, over a million tons of dust entered the atmosphere in one day, it would probably cause issues. Also, if instead you convert it into a number of smaller rocks, the area of affect can be greatly increased. Instead of one large event, you might have a dozen smaller events, but overall increase the area of destruction.
Bottom line is, the best thing is to get the entire rock out. If you can't do that, then unless you have enough time to allow for a wide separation of  the rocks, on the order of a million or more miles, you are probably better off just letting the rock hit the Earth intact. If you have the opportunity to do something tens or even a hundred years in advance, then trying to get it to disperse might be feasible, but otherwise, moving the thing completely at once is your best bet.

