Aging is the same cell and tissue damage that causes the age-relates diseases you mention. The only difference between aging and disease is how much damage you have, how far the cascade of cause and effect and secondary consequences has progressed. "Aging" is subclinical "age-related disease". There are few root cause forms of damage, but the cascade is as complex as our biochemistry, so there are any number of fatal endpoints and horrible declines along the way.
The only plausible near-future way to remove age-related disease is to periodically repair the damage that causes aging. Nothing else is going to work.
Consider this when thinking about living life to the fullest all the way up until death: if you are so healthy (meaning lacking in damage) that you don't suffer age-related disease, what are you going to die from? Your mortality risk is a function of the load of cell and tissue damage you currently bear. The only way to prevent age-related disease is to have a low load of damage, and consequently a low rate of mortality.
Given today's accident and murder rates, an ageless human undergoing comprehensive damage repair every decade or so would live for 1000 years or more before encountering a fatal happenstance. If you live much past a century from now, I'd have to imagine other technological options opening up that make that number seems small.
On overpopulation, it seems pretty clear from the research carried out on this topic that agelessness will not result in problematically large populations. Wealthy regions are already declining in population growth, and poor regions decline as they become wealthier. The poor are being lifted up rapidly.
See, for example, Demographic Consequences of Defeating Aging: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3192186/
It also seems fairly clear that what people call overpopulation now would be better described as some mix of kleptocracy, war, other bad governance, and endemic poverty that in many cases is being addressed at a fast rate in comparison to past history. Given the long list of regions that have lifted themselves up from peasant farming to technology industries and wealth since the 1950s, one has to look at those that haven't succeeded in this, and ask why: it certainly doesn't correlate with number of people.