I notice that your definition of 'cut to the quick' is "If someone is cut to the quick by something they are hurt and upset". Perhaps 'deeply wounded' says it better.
My reasoning is that quickthorn is an alternative name for hawthorn. If planted the right way up a branch of quickthorn will produce roots and grow into a new hedgerow plant.
Couch grass is also known as Quick in some parts of the country (England) and is a weed difficult to eradicate. In times past labourers were employed to 'pick quicks'. I imagine this refers to the fact that any fragment of cylindrical round root of this grass, if left in the soil, will regenerate into a complete plant and it is advisable to remove as much as possible.
From this I conclude that the original meaning of quick must have been a reference to what we would now call undifferentiated tissue. That is tissue from which parts of the body can be generated.
To cut to the quick is to cause more injury than a hurt or upset it is more life threatening than that.
Whilst the meaning of words changes it is as well to remember their original meanings
