Never take the findings of one study as proof of anything.
This is an observational study which took data from a patient database looking for an association between ADs and cardiovascular disease and all case mortality. The problem is anxiety/depression/stress are themselves by far the leading cause of premature death which makes it difficult in this case to determine whether the cause of a death was the ADs or the disorder it was being taken for and the disorder's impact on health. The authors have tried to account for many of the potential non med causes, but there is a lot of guessing involved. Change some of the confounding adjustments even slightly and the results could be vastly different.
There are also a number of other issues, including:
The dataset did not include dosing information, so this had to be estimated.
Adverse events were assigned to a med from when the first prescription was filled so if the patient had a heart attack the next day responsibility was assigned to the AD. So how did they determine whether the med was actually taken straight away? Ime, many just stare at the box for some days, even weeks, before taking the first dose and some never take it at all, but claim they did.
They didn't have enough patients in the selected cohort to compare outcomes in short and long term med use.
One indication that the study results need to be treated with considerable caution is that it found that SSRIs reduce two of the leading cardiovascular risk factors, high BP and diabetes onset, yet supposedly they increase adverse events??
The authors themselves say that the results need to be taken with caution. From the study's 'Clinical implications' section:
Since this is an observational study, our findings do not imply causality, and highlight the importance of further work to investigate and elucidate potential mechanisms. [my emphasis]
I also urge you to read the study and especially the "Strengths and limitations" section.