Is it feasible to use anchovies to improve the taste of random dishes?

If you really don't like anchovies at all, the others who have said "don't use anchovies if you don't like them" are right. Don't use them.

But if what you object to is a bite of food that just tastes like anchovies, you can certainly avoid that. Just use them spread evenly through a dish. For example, if they're minced and mixed in during cooking, the flavor will be well-distributed, giving a more mild umami and fishy flavor. If the dish is something the anchovies go well with, this might work out for you. And since the flavor is mixed in, you can use more or less as desired, while if you're putting sizeable chunks of anchovy filet on a pizza, there are always going to be bites that'll have a lot of anchovy flavor.

So, random dishes? Probably not. But things that they go well with, sure. For example, a lot of people who wouldn't like anchovy pizza will like greens braised or sauteed with garlic, anchovy, and parmesan.

(That said, my grandmother's anchovy pizza just has little bits of anchovy pressed into the dough, before anything else is put on top. The flavor spreads a bit; it isn't at all overwhelming but it is noticeable and good. And the people who really like anchovies get to eat the rest of the tin.)

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.