I think I am very a very limited creature, because I really don't get the warlord appeal.
The class really has its origin in 4th edition, where each class had a defined power source, and a defined role. A balanced party would have at least one of each role before doubling up on some of them, but the actual power sources didn't matter so much. The power sources in the core rules were Martial (pure non-magic skill), Divine (drawing on the power of the gods), and Arcane (magic); and the roles were Defender ("tank" in MMO parlance - good at taking damage and punishing those who attack the defender's allies instead of the defender themself), Striker (single-target damage dealer), Controller (debuffer and area damage dealer), and Leader (buffer and healer). So the fighter was a Martial Defender and the paladin was a Divine Defender - they both served a similar role in the party, but using different methods.
Similarly, the Warlord was a Martial Leader - a counterpart to the Cleric's Divine Leader. Both had the job of healing the party and making the party's attacks stronger, but they used different methods. The warlord used Inspiring Word for his healing, and either used powers with a tactical bent to make sure all the party members were in optimal position for dealing damage, or powers based on inspiring presence to spur party members to greater deeds. A typical attack for both the Warlord and the Cleric would be a variant on "deal some damage, and an ally gets a bonus to their next attack against that target."
The appeal is that many people want a character that can serve the same role in the party as a cleric - healing and buffing - without magic. They don't want the cleric, which is the most D&D-specific of all classes, to be a necessity when going adventuring.