A few things:

1. A stroke is not death. A stroke is an either a blockage of supplying arteries to the brain (ischaemic stroke) or a rupture of (most often) an aneurysm of one other arteries (haemorrhagic stroke). If left untreated, part of the brain dies off, but that is entirely survivable. Old people can have multiple holes in their brains from minor strokes and still walk around. If a large area is affected people may become hemiplegic, but a stroke generally doesn’t cause death.
2. A person generally doesn’t just stop breathing with a functioning heart and brain. A rare exception could be a basilar artery occlusion or haemorrhagic stroke affecting the midbrain.
3. We don’t really have a lot of experience with artificial hearts that provide pulsatile flow through pneumatic pumps, the few that have been implanted were mainly for research purposes. What we do have experience with is LVADs, or left ventricular assist devices. These are rotating centrifugal pumps that are implanted inside the heart and aorta and provide laminar, continuous flow. These pumps are so strong they can keep the brain oxygenated even if the heart itself stops pumping (e.g during ventricular fibrillation).
4. As for your question: death would be brain death or multi-organ failure. The blood would still flow and provide oxygen, but the end organs (including the brain) would not have enough functioning tissue to sustain life.