The Warlord is most definitely support. That is its primary archetypal role. Leadership, however, comes from the player/character and not the class.
Support is a
part of its archetypal role. It's also very explicitly labeled a Leader.
Leadership is an explicit part of the concept, and one we've been working on modeling in the Warlording the Fighter thread with a very real mechanical effect.
Sorry if that bothers you...
I cut some out, namely all the real life stuff that does not pertain to the Soldier background in the least.
Why is that? Did understanding of and focused training in Leadership not exist until modern times?
Does real-life not inform the game?
I didn't realize that the concept of a Soldier originated in and was defined by the fantasy genre...
There are several portions of the Soldier background that are quite implicit about the fact that you can be a 1st level sergeant, general, or other commanding officer, and explicit about the fact that you do have rank and authority.
I didn't say otherwise. You however implied that the Soldier background automatically meant Sergeant or General.
("Congratulations, you are now 'an apprentice general or level 1 sergeant.'")
As I said, that is Wrong.
You are correct that rank and authority are explicit in the background; but what rank you get is not. Low rank is rank also, and Authority is not the same thing as Leadership. Conflating authority with leadership is a common mistake; a mistake that leads many to not want a Warlord class, falsely believing it would be a character predicated upon telling everybody else what to do.
Simply gaining an understanding of what Leadership is can alleviate that error in perception.
So, as I said before, there is no narrative or mechanical expression of Leadership in the Soldier background.
A 1st Level character being a Sergeant or General, or anything other than Basic Soldier/Private, would be an inconsistent narrative. There's nothing in the rules saying one can't, but the rules also don't protect one from running a game in a silly, inconsistent manner.
Imagine how silly
The Empire Strikes Back would have been if Luke and Han were Generals instead of Captains a mere three years after the Battle of Yavin.
General Soldier Fluff: "When you choose this background, work with your DM to determine which military organization you were a part of, how far through its ranks you progressed, and what kind of experiences you had during your military career."
Specialty: Roll a d8. If you roll a 1, then you are - as per the chart - an officer with an official military rank. Congratulations you have received leadership training prior to level one. You can be a sergeant by level one. You can be a lieutenant by level one. You could even be a sergeant with any of the other rolls, since a 'sergeant' is not an officer, and your rank is mostly determined at level one.
Military Rank: "You have a military rank from your career as a soldier. Soldiers loyal to your former military organization still recognize your authority and influence, and they defer to you if they are of a lower rank. You can invoke your rank to exert influence over other soldiers and requisition simple equipment or horses for temporary use. You can also usually gain access to friendly military encampments and fortresses where your rank is recognized."
The chart doesn't list sergeant, and officer is not a rank.
And again, I didn't say a DM couldn't allow you to be a Sergeant or General at 1st level. The game's rules do not protect anyone from creating a silly or inconsistent play environment.
Also, ask any 2nd Lieutenant how much authority they actually have.

Most would tell you they're the OIC of Paperwork and Unit Functions (Thanksgiving/Christmas party, Ceremonies, Open Houses, VIP visits, etc...)
And the less it fits for me.
You have my sincere sympathies.
While still coming with all the same martial baggage, because the Fighter Warlord (NOT Fighter Leader) will be debated into the ground about the nature of any healing they may use, as many warlord advocates would still want martial healing. And you will equally find opposition to the Warlord PrC if it imposes its inspiration/will upon other players without magic.
Yeah, I'm not buying this argument any more.
If someone has a problem with inspirational recovery at this point, they're never going to be a fan of the Warlord. Period. Simple immutable fact.
The Warlord class is for those who
do want a Warlord, not for those that don't. Therefore, inability to understand or accept inspirational recovery is not pertinent to the discussion of how to present a Warlord. I'll certainly discuss it with these people, if they're interested in discussion, and do so in an attempt to change their minds, but it will not inform for me the direction of the class design.
Frankly, being bothered by inspirational recovery because of choosing to hold a view of it as being imposed upon their characters is a specious argument. It's no more imposed on a character than magical healing by a Cleric is. If one has a problem with that aspect of Clerics,
then they have a valid argument. If they don't, then no.
And inspirational recovery does not carry martial baggage, that is unless you consider the Bard a martial class... (excepting 4E...)
Jolly for you. You are free to pursue that preference so long as you do not actively attempt to sabotage or argue against the desire of those who want it expressed as a class.
No. Just No.
If you don't understand why, I suggest you ask a moderator.