Like I said, I think any time you go outside of the player character mechanics describing the character himself, I think you're on thin ice.
I know you said this, but I still don't understand why? What is the risk? What are you envisaging going wrong?
Because those types of mechanics can cause headaches for the DM.
Well, presumably the group has already agreed to a certain system before they start, so this has been dealt with.
What to do if his adventure setting doesn't allow a mount-animal companion/etc. to come along? What to do if the political situation in the world doesn't make followers practical?
In systems in which status can be purchased as part of PC build, the GM is generally expected to shape the ingame situation to reflect those choices. Just as, in classic D&D, if a player wants to play a paladin, the GM doesn't present, as the first scenario, an orphanage massacre. (Three Days to Kill, an early d20 adventure, has a more subtle discussion of this sort of issue, to which it potentially gives rise.)
Conversely, that a system both (i) builds a mount into a PC's advancement, and (ii) doesn't give the GM any advice on how to handle this, is a weakness in the system.
What to do if the DM wants to change the magic item frequency in his campaign, but those items are assumed parts of character advancement that differentially affect different character types?
Again, the GM can't. Not without changing the PC build rules. Just as a 3E GM can't change the frequency of scrolls without changing the PC build rules that give all wizards Scribe Scrolls for free.
the "assumed" wealth of a high-level D&D character is quite an extreme imposition on the game world; arguably an even greater one.
I'm not entirely sure what you have in mind here, but obviously a system in which certain benefits are purchased using PC build resources has to be careful about handing out other ways of acquiring the same benefits: be that items, feats, skill points, or whatever (contrast 3E, for example, with RuneQuest or Burning Wheel: in the latter two games you can improve skills by adventuring, or by paying for training; in 3E, you can only improve skills by adventuring - presumably 3E is limited in this way for balance reasons).
Should the player of King Arthur the character be able to declare his character a King as well?
Maybe, maybe not. That depends on how the game is set up. It doesn't follow that, just because the game supports items as part of the PC's build it supports status, nor vice versa. The Dying Earth permits items but not status. HeroWars/Quest likewise. As you yourself note, classic D&D permits status but not items. (As well as the rule you mention, there are the rules in AD&D Oriental Adventures for a samurai PC becoming an estate overseer at (I think) 5th level, and a provincial governor at (I think) 7th level.)
HARP permits both (although I wouldn't say that HARP is an especially good example of how to do either - it doesn't have the requisite GM advice on scene framing or adjudication of consequences to support them).