From what I've seen, numbers citing the "Cost of obesity" include both the healthcare spending and the lost work.
There's lots of addition, but no subtraction.
If obesity is shortening lifespans, shouldn't the costs of social security and the (expensive!) late life health care be deducted for the years they've taken off?
I think the specific point to be made in terms of the obesity health care costs link is that the most expensive thing of all is chronic illness—things like diabetes that don't kill you if treated properly but that require a lot of care. And obesity is thought to cause diabetes. So there.
But I agree with the general point. Especially if we're talking about direct fiscal cost, the mere fact that getting sick is expensive is no particular reason to think that better health would save money. On a long enough timeline, the survival rate for everyone drops to zero. Staying physically healthy long enough to experience major mental deterioration may well be the more expensive option. The best reason to promote better public health is because healthier people are happier and live longer, and having happier longer-lived people is a good goal to pursue.