They wouldn't be us. We will still die when our bodies fail. But maybe there will be some AI tricking our friends and family into thinking we're still there.
The body is constantly changing. We already know about physical and chemical abnormalities in the way your body works affects your "person" and we can sometimes address them with surgery or drugs. The physical body's limits impact the observed brain. If uploading is possible, if there are some examples of working cases, if I don't hurt anyone why not try it?
Is this the Ship of Theseus, or is it a slow but nonobvious death?