I thought you might be asking something deeper, but around these parts you never know. Years ago I met a man who was fond of saying, in each of us there are two. One being who we think we are, the ego driven one IMO, and the other our true self which takes more introspection than our fear based primitive mind can deal with. It is that wall of fear that must be breached to see ourselves for what we really are and then changes can be made to the ego driven self. So I guess to answer your question, do we exist when the mind is silent? Yes, I think so, but it is our true self that exists and not in a physical way. Only the ego driven self has a physical presence. 'We" are not here. We are everywhere.
So we know the Buddhists say that the root of suffering is desire, but without desire how can life go forward? To answer desire is to answer nature's call. Nature wants us to enjoy ourselves does it not? The rock does not desire, nor does it flower.
Buddha also said that the true root of human suffering is our absorption with self. He offered the way out of that suffering through thinking and doing for others first. Nice concept, seldom practiced. Writing a check is not what he was talking about.