I'm not too worried about having chunks go missing. The moon has enough gravity to hold itself in in a sphere-ish shape. The mines will collapse in on themselves long before a huge chunk goes missing. (Plus, if we have things that can do that kind of damage to the moon, damage to the moon will be the least of our worries.)
As for knocking it out of orbit, the moon weighs 7.35*10^22 kg. The largest nuclear weapon ever detonated, Tsar Bomba, had a yield of 2.1*10^17 J. So, uh, not too worried about it. We're a few orders of magnitude off on that.
There's some legitimate concern about scarring the face of the moon for future generations. Yes, the moon is really big and really far away such that you probably wouldn't notice even a very large mine with the naked eye, but telescopes are cheap and plentiful. Possibly something you could deal with by treaties limiting the size of mines to less than utterly huge? At least there are no indigenous people to worry about. (Or, as history has demonstrated, not worry about.)
From reading the article, it looks less like "getting a permission slip from NASA" and more like "agreeing to cooperate with NASA." This seems reasonable to me, for US companies. Obviously, if commercial space exploration takes off, we're going to need some international agreements in place.