Your question made me examine this particular prejudice I have. Goes back to the 80s when my sophisticated, foodie boyfriend got me my first Melitta and it was different than my parent's Mr. Coffee. So I have stuck with it. Cones are not new, I recently saw a scene from the old Mary Tyler Moore show and she was carrying a tray with a Chemex on it - a method that seems to be enjoying a comeback, if Amazon reviews mean anything. I enjoy my coffee black with a lighter roast, I want to it to have a sour, citrus taste or I feel cheated. If it does not have this it tastes flat and boring to me. I am definitely not a "super taster," in that I prefer black coffee, dark chocolate and any kind of intense flavors. This profile seems to go with the greater extraction that occurs in a cone.
$1 is a 2019 article that references a study conducted on the cone/basket question. After you read it, you may conclude that a basket gives you the flavor that you want. That has an advantage because basket filters are cheaper and machines that use them are more readily available. Edit: I just saw that you use a French Press. I would never turn down a cup from one, they are fun to use camping too.