As I just wrote, nothing is infallible. However, if it was just the chuck that became live, one would be very hard-pressed to earth it, even if one wanted to.I recall a 240v hammer drill that my father owned (branded under one of the big shed's ranges) that when it expired managaged to put 240v (or at least some fraction of) onto the metal parts of the chuck, dispite it being supposedly class 2.
I don't think that anyone doubts (or has suggested otherwise) that the ideal is to have no touchable metal parts, coupled with adequate ('Class II') protection against contact with any live parts. However, you have illustrated one problem with that yourself - although I have seen plastic chucks, they are not particularly realistic. Nor do I personally believe that one should earth (or attempt to earth, when it's a moving part!) any touchable metal of a reputable Class II item - but, as I have said, I am not totally happy with an instruction that such metal "must not" be earthed.However is the risk of the odd failure in this manner less of a risk than the old class 1 metal drills, that you might be using outside holding tightly onto the case, possibly on a TNCS earth, possibly with no RCD where a fault would put a voltage onto the that case you are holding for upto 0.4 seconds..... much longer if the earth happened to be broken/not connected ....
Kind Regards, John