The core of truth on the base of this tale is tiny. Certainly not large enough to warrant emulation.
Anchovies are a source of concentrated umami flavor, and people tend to like umami flavor, but most of them don't recognize its presence. So, if you were to add a little bit of umami to food, it is likely that many people will find it well seasoned.
But anchovies are really not the best source of umami to add to food. They just contain too much of other flavors (salty, frequently also sour) and aromas (fish aroma). It is like hearing that most people like the taste of sweetness and start adding a bit of beetroot juice everywhere, because it has a high sugar content.
There is no way to add anchovies without adding fish taste. You can use tiniest amounts, but then of course the effect of the umami would be tiny. If you add more, it will taste like anchovies, there is no way around that.
If you really want to have more umami everywhere, you should start adding pure MSG. This is similar to adding sugar to foods, and won't result in off tastes.
All in all, the story may have some teaching value as a parable, but following it in real life is akin to saving a mad-with-pain lion from a trap with your bare hands. Leave it to story heroes.
Update Many disagreeing comments mention one or two foods which do get better with anchovies. I didn't bother writing about that exception, because the question was whether you should use anchovies everywhere, and my argument was that they are like every other food: there are a few places where they fit well, but are not a use-it-everywhere spice like salt.
If your food is already very high in umami, and also has strong aromas, anchovies will fit well there. The reason is that people already expect the umami there, and when it is stronger than expected, it is perceived as more pleasant, more tasty food. A small amount of anchovies is enough to emphasize the already present umami taste, and the other aromas cover the slight fish aroma.
But you cannot generalize from that exception class to other foods, just like you can't say that, because lemon zest makes yellow cake taste better, we should start putting lemon zest in all our foods. So I disagree when a long list of comments says that my answer is not true because beef/tomatoes/mushrooms/whatever high-umami-rich-taste stew tastes even richer with anchovies.