In which case why are you starting with a full caster base?
Because although the bard conceptually doesn't fit (it uses magic), thematically (leader that can fight but main strength lies in improving others) and mechanically (abilities boost companions in a number of ways) it is a match for the warlord.
Unlike the fighter, the power of a warlord does not lie in their personal damage potential, but by how they improve their companion's power. Thus the abilities rather than attack rolls are the focus of the class - just like a full caster.
You can't do what you want without making the Warlord flat out OP compared to Fighters. It's a terrible starting point for a balanced homebrew class.
Why compared with a Fighter? Its a closer match to the Bard. In fact, why not compare to all the classes rather than just one?
You asked why both can't exist. Let me try to explain.
Under the current classes, if my concept is elite warrior that is a great leader I pick Fighter/Battlemaster and pick the appropriate manuevers and I'm both a top tier warrior and a top tier leader. Now suppose we created a warlord class with much better leader mechanics and minimal warrior mechanics. I no longer can play a class that is an elite warrior and a great leader. I must choose elite warrior and average leader or great leader and average warrior.
Because you're
not "both a top tier warrior and a top tier leader". You're a top-tier warrior: 5 out of 5 certainly.
However Battlemasters are hardly "top tier leaders". They're 2/5, maybe 3 if they're willing to drop their warriorness tier.
Wizards are probably 3/5, clerics a 4, and the top-tier leader class would be bards at 5/5.
Hence the wish for a martial Warrior 2 or 3 and Leader 4 or 5, which cannot currently be achieved.
Do it ina new game or with a total class rewrite. You can’t do it with a single new class or you ruin the balance the game does have.
As long as the class is less powerful than the most powerful other choices, its not breaking any balance.
IMO. If the Warlord is a Martial Class it should be comparable to the other Martial Classes.
That would depend on where you view the upper bounds of what leadership is capable of. I like leadership abilities but I'm not sure that full leader in most settings should be a competitive option vs full warrior or full mage.
I get that but whatever you produce should still be roughly balanced with the fighter/barbarian/paladin/rogue/monk/ranger. So it's not like you are actually free from such 'flaws' even in this case.
A new class should be less powerful than the most powerful class, and ideally around the mid-point of all the D&D classes. There is no particular reason to try to aim at the mid-point of a rather arbitrary subdivision. Particularly when thematically the class does not fit into that division.
You're describing combat ability, though, not exploration and social ability.
Is there a particular reason that, having established that Fighters and Rogues are extraordinary, we should start demanding "But only in combat".