It's a question of natural authority vs. imposed authority.
The Cole incident took place on Chelsea property, and Chelsea pay the wages of Cole and the kid who got shot. Chelsea pay Ancelotti's wages and Ancelotti decides if Cole takes the field.
In this case sanctions would be intended almost entirely as a deterrent. Chelsea will accomplish far more than the FA could if they "handle" the matter internally, because while the FA can suspend Cole for three or four matches, Chelsea can hang him out to dry for a season - all they have to do is decide they don't like the cut of his jib.
An issue arises when a player's quality means his club are loath to discipline him, or when the club simply does not handle issues of discipline overall.
The other issue is that there's no clear relationship between a club's success and its internal harmony. In fact if anything clubs with an internal culture characterised by punitive aggression tend to be the most successful.
A system in which the club decided on an internal sanction but maintained open communication with the Association would be preferable.
