I would argue the 1e Fighting Man was closer to a Warlord than to a 5e Fighter. They grew to be leaders of mundane men and commanded cohorts. As domain building and underlings fell from favour, the Fighter lost all that leading ability, leaving only 'I swing my sword' behind. The Warlord is just a return to those roots, but this time synergizing with the other PCs.
Instead of having a Warlord class itself, I wouldn't mind the Fighter being improved and fleshed out by more Warlord traits as part of their base class features. A Fighter would not longer be seen as just a town guard or a random soldier, they would be Sergeants and Captains and could make use of their commanding presence and military education in social situations. Reclaim the old status of Lord!