Sure, but first a fighter has to get persuasion - not too hard, but it's not on their skill list. That means getting a race that can provide it or a background that can - which may not be optimal. And if the fighter "just" takes the skill they'll be pretty bad at it. Basically their role will be to assist the guy who's good in the party (to give them advantage). Assisting is all well and good, but, sometimes, you want to be the actual guy!
Even the current option in Tasha's - take commanding presence to be able to add a superiority die to (some) charisma checks is a massive investment as fighters don't get all that many maneuvers (or dice for that matter as every use will impact their next potential combat).