As others have noted, the notion of "fundamental flaw" is capable of different interpretations.
I think that there are five key aspects of 3E that the designers have identified as in need of repair:
*Significant parts of the rules are not being used (or not used with pleasure) by a majority of the players of the game.
This would include (for example) the grappling rules, and the elemental planes.
This sort of problem is not caused by supplements and rules options - in fact, it can be fixed by them.
*Significant parts of the character build mechanics are hard to use well and/or a prone to misuse or breakage due to unforseen synergies.
Examples of the first would include multi-classing with spellcasters. Examples of the second would include CoDzilla (though maybe some of that was foreseen, and offered as a carrot to have people play the healer). The role of magic items and "Christmas trees" in character build also fall under this heading.
Supplements can exacerbate this problem (by creating more unforseen synergies) or ameliorate it to an extent (by giving us more Mystic Theurges and Practiced Spellcasters, or new options for items as in the Magic Items Compendium). But in many ways it is better tackled from the ground up (eg the way that Conan tackles the magic item issue with its stat gain rules).
*Significant parts of the rules lead, due to the overall design logic, to an unsatisfactory play experience.
Examples include those part of the spell rules that lead either to players of wizards frequently doing nothing, or to nova-ing and overshadowing non-spell-users.
As far as I can see, this does go to the fundamentals of 3E mechanics. Solving it requires either:
(i) Changing the overall design logic by going back to 1st-ed assumptions about play - in particular, that the crux of the play experience will involve operational planning and GM-mediated solutions to non-mechanically expressed challenges; or,
(ii) Changing the suites of class abilities so that they work better with the present design logic - in particular, that the crux of the play experience will involve character build mechanics, and the overcoming of mechanically-expressed challenges by clever applicaiton of the action resolution mechanics
Of RPGs that I'm familiar with, AD&D is the only one in which option (i) has ever been prominent, and the general trend of the game over the past 25 years has been away from it and towards (ii). Solution (ii) also has the advantage of addressing the second point above.
The problem doesn't arise from supplements, and it can't be solved by them either. Fundamental change is needed.
*The game is very hard to GM, especially for high level play, because of the complexity of monster and NPC build rules (eg skill points) and the complexity of monster and NPC action resolution rules (eg the many spell-like abilities that high level monsters often have).
Common build rules for PCs and NPCs/monsters is pretty fundamental to 3E, and is a flaw: what GMs are looking for in their play experience is quite different from what players are looking for (eg complexity of PC build is a plus for many players, but complexity of NPC/monster build is a minus for many GMs). Fixing these rules is not something that can be done via options and supplements.
*Currently, some aspects of trope and flavour are seen as impediments to easy uptake of the game, and particularly to easy GMing.
These include sometimes obscure distinctions between monsters (eg demons and devils), and the perceived need to detail a world before the PCs can interact with it. The 4e designers are therefore changing those things (eg succubi become devils, "points of light" becomes the default setting).
These are pretty clearly not fundamental flaws.
Anyway, that's my take on it.