There's only one thing I add to my coffee: hot water to the fresh grounds when I brew it. I want to taste the coffee, the roast, the sweetness, the acidity, and all the different elements from where it was cultivated, how it was cultivated, and how it was processed. Adding sugar, cream, chicory, or anything else covers up what the coffee brings. Looking back, when I was adding stuff to coffee, I was covering up my poorly understood brew methods.
Now, if if I was inclined to add chicory, it would be to a fairly flat neutral medium-dark to dark roast that doesn't have much going on. That's so that I could get a clear taste what the chicory is bringing. But I have as much interest in chicory as I have in other flavored coffees. Which is when I'm feeling morbidly curious.
Now, if you want to add it, you do you. But there is a lot of depth to coffee when you start digging into using good beans with different brew methods, brew recipes, along with that search for the "perfect cup" of coffee.
If you are interested in getting a good cup out of french press, James Hoffmann has a good recipe. It's basically a modified cupping recipe.