The "conventional" D&D approach is that the character affects the game world, and the player controls the character, and has absolutely no other influence.
This may be the conventional approach for 3E - I wouldn't really know. It's not the conventional approach for Gygax's AD&D, though - the player by rolling a saving throw can affect the gameworld without controlling his/her PC (for instance, Gygax explains that a successful save against dragon breath by a fighter chained to a rock might mean that the fighter noticed a cleft in the rock and ducked into it).
Also, in classic D&D the player, by having his/her PC collect treasure, can make his/her PC more capable in the gameworld as a direct metagame benefit. That is not an infuence on the gameworld causally resulting from the actions taken by the PC within the fiction (eg no one, including Gygax, supposes that collecting treasure is an ingame cause of increased prowess).
The point of metagame mechanics is to give players the ability to affect the game world, distinct from the abilities of their character.
Yes. Classic D&D has them (saving throws, XP and I would argue hit points are some of the key ones). 4e has them. I'm not competent to judge on 3E, but perhaps it has fewer of them (I don't really undertand its XP system, though at first blush it seems an odd mix of ingame and metagame; likewise its hp; its saving throws are obviously very different from classic D&D saving throws).
Thus, a true metagame mechanic has nothing to do with the character. It is not dependent on what class or level the character is, and the character is not aware of or in control of them. That's why I gave the plot point example.
The character doesn't spend the resource; the player does. But it is a non-sequitur to argue from this to the conclusion that metagame resources cannot be differing parts of PC build from character to character. In fact, in RPG design t's utterly ubiquitous for metagame resources to be built into particular class choices (for instance) for balance purposes. That's why classic D&D fighters have more hp and robust saving throws. That's why thieves in classic D&D have a more generous XP table.
You've mentioned Cortex a couple of time: Marvel Heroic RP very obviously uses metagame resources to balance across different PC builds (lower dice roll proportinately more 1s and hence generate more Plot Points, to balance for the lower totals and lower effect dice).
Burning Wheel permits players to spend build resources on Traits that earn metagame points (Persona Points, which can be spent for bonus dice or death avoidance). Monte Cook's Arcana Unearthed has the same thing, where a player can use build resources (in that case, feat slots) to acquire a feat that will accrue metgame points (Hero Points, I think they're called).
What you have in 4e is some conventional, some metagame, and no clear distinction between the two. One power might be clearly metagame (i.e. something that the character cannot do), and another might be an ability of the character. Thus, no one really knows what a lot of the rules even mean. Certainly, for diehard fans its easy enough to rationalize these mixed in-game/metagame rules, but that lack of transparency and all the consequences thereof are kind of a problem for the rest of us.
In my view this is a distinctive and appealing feature of D&D. You have mechanics which can be treated as metagame or ingame from moment to moment of narration. Hit points and (non-3E) saving throws are time-honoured examples. 4e adds skills and some powers to that list.
Marvel Heroic RP does the same things, interestingly. For instance, during a Transition Scene Plot Points can be spent to activate a Specialty (the closest thing that game has to skills) so as to produce a Resource (a bonus die for future dice pools). Is spending the Plot Point a player action only, or a PC action too? That seems to me to depend completely on how it is narrated at the table. Suppose I (a player) spend a Plot Point to activate my Covert Specialty in order to get access to a passkey that will get me into the secret HQ. I narrate this as me scrounging around, digging up my old contacts, and eventually persuading one of them to make me a fake passkey. In that narration, it seems to me that spending the Plot Point is not just something the player did, but also something the character did: expenditure of the Plot Point correlates to effort spent within the fiction by the PC to acquire the Resource. But another narration might be "As I'm walking home in my secret identity, I notice a wallet lying on the ground. I pick it up, and luckily within it I find a passkey belonging to a low-level employee of secret HQ." In that case, spending the Plot Point is clearly happening at the player level only.
Neither narration is obviously preferable. Each makes sense within the fiction, and each gives the GM opportunities to develop future complications (fake passkey vs stolen passkey).
The 4e warlord is versatile in the same way. Consider a power that grants an ally a bonus action. This can be narrated as taking place ingame ("The inspired PC pushes harder, urged on by the valiant commander.") It can be narrated as purely metagame - this is how some lazylords play. And some PCs might flip back and forth - eg the "princess" warlord might sometimes be pure metagame, but when she screams for help or issues a biting rebuke then the bonus action actually has the PC's action as its ingame cause.
I guess I don't see what "all the consequences" are that are "a problem for the rest of us". If you're happy with metagame abilities you're already happy with introducing material into the fiction that did not have, as its fictional cuase, any PC's action. With 4e you just keep doing that when the inclination strikes you.