The idea behind all skills is implementing them in your daily life. If you are a statistician who never really uses math in daily life, then it's not the math at fault, but you simply haven't found a way to make it a habbit yet. This is also why the points in the article are rather broad and not really statistics related. For instance, how do you decide when which details are important and when the big picture is important (two points that follow each other directly after one another). You'll probably say gut feeling. But if you don't use the science to consistently train your gut, then your gut feeling isn't that much better than mine (I'm no statistician).

Yes there is a point where you get so much into the principles that you don't need to calculate everything anymore. However, that point is not after graduation, but after doing more math than the people around you for about 3-5 years, maybe 10 for some. And a huge part of your intuition would probably be around not needing a calculator to put the numbers together, and about finding many different solutions to the same problem, understanding that each has their pros and cons (e.g., different fittings that all tell you something about your data cloud).

If you answer a broad question with "it depends", it's a good sign you only think you are well trained (happens to all of us all the time). If you think "it depends, if A then B but Z, if C then D but Y, if E then G but X, ..." then you are probably well trained.