Although very fine filtering provides a cleaner lighter coffee, a certain amount of (very) fine grounds is needed in the coffee to give it the right texture (IMO).
I used to use a Moka pot, for years - that was my first introduction to "real" coffee, and it is the way my French family have made their coffee since "time immemorial"...(and certainly since well before my birth!)
My own path went from Moka pot to espressor machine, to French Press / Plunger. In the end I prefer the French Press for the type of beans I like roasted the way I like them. I grind fairly fine (not TOO fine) - too coarse and it doesn't get enough extraction of flavour, too fine, and you get too much in the way of solids in the coffee.... I am content with my perfect compromise, which provides just enough solids in suspension, to adjust the mouthfeel.