The advantage of calorie-counting is that it's quite easy - especially when you use something like My Fitness Pal. That doesn't mean it's correct, of course.
I only skimmed the article but it doesn't really advocate an alternative. It highlights that the person they were following "eat real food, not food ‘products’" and that weight watchers uses points instead of calories. So it doesn't seem like there is a nice actionable item that someone could take away from the article and apply to their own life.
I don't think the article really advocates any alternative, but I think things the best benefit of MyFitnessPal is to promote awareness of what you are intaking.
I have been receiving health coaching for last 4 months or so, and most important part that working for me is to be able to be conscious about food choices, and also being able to reflect what I have been eating and doing how well I'm doing with sugar, carbs, protein, fiber, etc., rather than just looking at the calories count. (and I do eat unhealthy stuff time to time, and I don't think my health coach is necessarily against that -- unless that's my regular habit, that is. It's more important to know how they balance out in the overall intake,)
It never occurred to me before I start measuring what I eat using MyFitnessPal. But now, seeing those numbers, I tend to consciously adjust my intake, for example, I will just eat 1/5 of that chips I wanted to eat, because I had bit of carbs today, etc., etc.,
It's obviously not an exact science.
If you go with a daily reduction of 1000 kcal, then it doesn't matter if you have an error-margin of 300-500 kcal, you still lose weight.
If you only try to save 500 a day, and maybe are not too exact in your logging, you might not see an effect at all.