There's a pretty strong anecdotal history of various types of animals tasting different based on regional diets.
There is also a video about a foisgras producer that acheives lots of flavors in the liver, like you say, with a mix of plants and fruits that the geese have access too.
I don't believe that you can salt your birds with high salinity plants. Animals seek homeostasis; if they ate something salty, they would drink enough water to flush their bodies out to a normal range of saltiness.
Further, I think grass fed beef has a mildly grassy flavor, but grass is 100% of the beef's diet. A chicken eats a huge variety of things in the yard, but it's still only maybe 8-10% of their diet. The rest is made up in commercial feed or prepared diet, to support their accelerated growth.
Some of the more famous specialty-diet meats (milk-fed poussin, acorn-fed ham) are animals that have a longer growth period than your average chicken, so considering the amount of feed consumed it seems to me the math works out better that some flavor from the feed will emerge in the meat in an older animal.
That said, I maintain that my birds taste more "mineraly" than a grocery store bird. The fat tastes cleaner. But this is also due in part to a longer lifespan (12 weeks vs 6-8) and a pastured, free choice diet. They have rocks in their gizzards. Now that we are switching to a whole grain/fresh ground feed with actual recognizeable ingredients, I do expect a flavor difference because it represents a huge amount of the animal's diet.