I'm sure we both have a dictionary John. Perhaps instead of discussing independence we should consider impartiality, which is surely far more important.
Indeed, that's what matters - but, to be honest, when people talk about 'independent' is this context, I imagine that what they are usually really thinking about is 'impartial'. I suppose the point is that, whilst it is perfectly possible for a non-independent person to be impartial, there will 'always be doubts' in some people's minds - whereas if one is totally independent, then there is little of importance that one could be 'partial' about.
Kind Regards, John