There are a few odd things about this approach.
First, 20% of people in a healthy weight being previously overweight/obese? This seems high? Could there be reporting bias or other things here? I'm still trying to understand the full study, but it is odd that more people lost weight than gained weight, according to the numbers I'm seeing in it. Although that may be partially explained by the fact that it was limited to people 45 - 85 years old.
Second, the methodology of taking anyone who was previously overweight/obese and is now healthy (weight-wise) into their own group and then attributing increases in mortality purely to weight seem suspect? There are a lot of things that can cause you to lose weight and lead to increased mortality that aren't related to weight, and those would be captured in here.
The conclusions definitely make sense and "feel" right, at least directionally, I'm just a little confused about some of the methodology to get there.