Because Gygax and Arneson read the same books (conan, vance, john carter, tarzan, LotR, fafhrd, e.e. doc smith, etc.,) and agreed that that was how things were supposed to progress. Heroes naturally become rulers, and rulers naturally deal with different kinds of problems than heroes. Ergo, the ROLE in your ROLEplaying game is a moving target, and you (the players) are naturally supposed to agree and adapt accordingly. Gygax was rather vocal about it, iirc.
. . .
Look at earlier versions of D&D to see when all the Leadership stuff automatically kicked in (became a feat in 3.x). That's when your game play was supposed to "naturally mature" and move beyond the dungeon crawl. It became a kingdom crawl instead... followed by an empire crawl... followed by a planar crawl... and topped with ascension. The Birthright campaign setting even gave action types for possible actions taken on a large geographic scale.
Except . . .
I've read those stories too (except for ee doc smith), and . . .
That's not what happens.
Conan becomes a king and . . . promptly goes on a killing spree. Sure it is stopping a conspiracy against him, but does he defeat it by talking and hiring flunkies? No. He defeats it by schmoozing with a sweet young thing and caving in skulls.
John Carter becomes warlord of Barsoom. Does the next novel cover his diplomatic negotiations and social reforms in detail? No. It picks up with a relative running about committing mayhem. And the next. Then for variation another earthman shows up on Barsoom. Then John Carter reappears - committing more mayhem.
Tarzan? All of one chapter about how he has some problem running his estate, then the next 20 chapters he is back in his loincloth, smacking stuff silly.
Fafhrd and the Gray Mouser! No wait, they just keep going adventuring until they retire. Then, when they come back for a mid-life crisis they promptly get kidnapped and have to go on more adventures.
Lord of the Rings! Ends when it comes time to do the actual ruling, which is summed up in an appendix. Even Tolkien did not try and right an epic saga about kingdom administration! (Which is both expected and unexpected in its own way.)
Now technically Vance wrote two of the Dying Earth novels about a supreme wizard who sent a flunky out on his bidding. Of course the majority of the text of those two novels then follows the misadventures of said flunky. And the fourth novel was about a conclave of super-wizards and their infighting. But then Vance also used words like "quotidian", and copying that style, adapting the magic system, and incorporating ioun stones is pretty much all that the game gets from Vance.
Nowhere in any of them does anything even vaguely like the Companion or Birthright system for kingdom management appear. At most, you might consider the random events tables to be a vague nod to some kind of sandboxed campaign, with a die roll determining which adventure to play next rather than DM design and fiat, but that's about it.
So really no, the fiction does not even reflect that model. (Which would lead to a longer rant of mine about the flaws in trying to make the modern version of the game reflect Appendix N.)
As for Leadership as a feat in 3E, yes it is. And as a result, unless the DM starts to give out bonus feats for such background elements, character optimization is going to tank rather rapidly when it comes to getting the skills and feats needed to optimize building a kingdom. 3E has the additional "feat tax" of Landlord in the Stronghold Builder's Guide, the plethora of guild/organization related feats in later books, and a generally less-than-spectacular couple of rules sets for running a business or creating an affiliation. Pathfinder actually manages to be worse with their Kingdom Builder and Downtime rules in terms of non-optimized choices, although Pathfinder did manage to create an entire additional sub-system for intrigue - which of course requires even MORE resource expenditure to optimize at the cost of combat effectiveness.
So unless you are going to allow heavily rebuilds/retrains at "ruler level", with a "go back" option if the PCs have to do fieldwork themselves, it really is not going to work.
Well, or if you get really weird and build an entire campaign around multiple PCs per player, with the additional PCs as the cohorts/followers/hirelings of the main PCs, and some really gratuitous bonus feat distribution to the "rulers" (did someone say "bloodline"?), with the campaign flipping back and forth between leaders and followers, and the adventures representing different events (random or otherwise) needed to build/protect/expand the kingdom.
Which my players actually enjoyed quite a bit, but I wouldn't expect it is for everyone.
Again though, we have a situation where the system really does not support that kind of play without a significant amount of hacking to the point that you really are either playing another game entirely, or going with a truly divergent type of campaign that overwhelms the base system and assumptions.