LVADs are implanted in your chest. Blood doesn't leave your body, but do they have external batteries. You have a wire coming out of your chest (or neck/head, sometimes) which is a constant infection worry. AFAIK the only external pumps are like, countertop kind of things.
They're ideally a stopgap, but even still it's typically a massive change in quality of life. The left ventricle is what feeds your entire body blood; the right ventricle only feeds the lungs. When the left ventricle fails, blood backs up and starts overflowing into the lung space. It's exceptionally unpleasant. The right ventricle is usually in much better shape so an LVAD can be almost like having a functioning transplant except for the risk of infection.