The Economist article includes wide-ranging information on the relationship between age and happiness in various parts of the world. The information you've provided addresses a small portion of that (the percentage of the population on anti-depressants as they age).
A population can be both more happy with age and (as a percentage) on more anti-depressants. Some potential explanations:
1. As we age, we have greater access to healthcare (money/quality employment/government), so those needing anti-depressants are able to get them more regularly
2. Even with access to healthcare, there may be greater willingness to make changes in life and confront unhappiness as we age
3. Perhaps those who age are happier overall because they're on anti-depressants. The Economist article doesn't address this possibility.
I feel like the flaw in your argument may be that increased anti-depressant usage among older adults indicates increased depression as a person ages. That doesn't take into account other factors that might cause anti-depressant usage to rise with age.