When you read a study like this you want it to be wrong. That is, if you are over 50 yourself.
The attention grabbing result is that surgeons aged over 50 got worse results than those aged 35 to 50. How can we pick a hole in that? The first temptation is to note that the study was based in France and blame it all on garlic and red wine.
Resisting that temptation, we can move on to methodology. Actually, the paper seems quite strong on that. A research question focussed on a well defined operation, with prospective data collection on surgical complications, and good compliance.
One possible explanation for older surgeons doing less well might be failure to adopt newer techniques but the authors tell us that thyroid surgery really hasn't changed much over the past few decades.
The relationship between age of surgeon and complication rate is assessed through logistic regression to deal with confounding factors such as case mix. Often papers using these sophisticated techniques give you no sight of the raw data, which makes it very difficult to get a feel for what is going on. This paper does not have that failing - there is a useful set of graphs with the raw data. These allow us to see that the whole 'worse over 50' bit depends on just three or four individuals, and in fact the oldest of them is better than his slightly younger colleagues.
What does it all mean? I'm not sure. But at a minimum it should prompt further studies which hopefully will be as well executed and reported as this one.