I also agree that there are a number of people involved in the care of a horse -- and that all of them should be responsible. However, - and this is where we may disagree - there has to be one person who has overall responsibility or oversight for the horse. Otherwise there is the potential for each of the parties providing pieces of the care to make decisions that are inconsistent with one another or worse contradict each other. The person with that overall responsibility is to my reading (and I am a lawyer) the person who in the truest sense has "care, custody, and control".
My point is that training in the truest sense requires involvement in and responsibility for (not the physical doing of tasks) in all aspects of the horses program. I am not sure we want a world in which trainers do the training ride and teach the client and know nothing else of the preparation or treatment of the horse (or are not responsible for decisions in that area). True horsemanship requires more than that - and so should responsibility.