It might fix the latter. The mechanisms involved in heart disease and diabetes are related to general aging as well.

Accidents and suicide could be handled too. Former by outlawing machines of murder (cars) and spare or artificial organs, latter by improving our understanding of neuroscience and psychology.

yes and no. Aging research wont, but that does not mean other science stops. Traffic accidents are at all time lows (in comparison to the amount of traffic) and will continue to do so (via automated driving, imidiate ambulance dispatching, etc.), homicide as well (maybe not in US but certainly in europe), and depression is under active research like being partly caused by the microbiome etc. Nothing is endless but getting 600 years old would still change a lot.