Depending on the edition, another major problem might be the prohibitive nature of the action economy disallowing anything resembling a balanced, and functional, simultaneous usage of sword and spell.
Not necessarily a problem though. After all, the action economy may be one of the things that naturally keeps the character from being radically overpowered. In some circumstances, 3.0 haste could allow very powerful combinations of attacking and spellcasting.
Another problem might be, due to the multiclassing system, the default stats of the character end up not being equal to the task of a front line melee combatant. Another issue (and not a small one) is the crippling effect on overall effectiveness of MAD.
Again, that's true, but I'd categorize that as a reasonable balancing factor.
Yet another problem, and not a small one, is (depending on edition) there may not be any thematic functionality that actually supports the archetype to make it distinctive; its just a pile of stuff.
I don't know. It's one of the few examples where something that isn't a class has a name that is not a real word and yet is recognizable ("gish"). The archetype is pretty clear. There are plenty of spells specifically to support it.
I think a large part of it may simply be the weird anathema between arcane magic and armor, and the need the designers have felt to dance around it and consider it as a balancing factor. After all, making a hybrid melee/divine caster has never been a problem; it's the default!
Wait what? Being able to create the character you want is a basic defining experience of all traditional RPGs, regardless of which one and which edition. The fact that 3.X offers a broader range of freedom in the mechanical end of this is not sufficiently important to many players to put up with the fact that most of those options are terrible unless you're a Wizard or CoDZilla.
That's wrong on two levels. First the premise is wrong. There's no evidence that any particular class or choice is that much different from another. The available spread of options has always been and remains a real and intriguing choice. Except possibly bards.
However, even if it wasn't, I'd say players would still want that choice. On the micro level, many players dive in to opportunities to spend character creation resources on things that are useless to fighting or even adventuring, simply to detail their character. In 2e, it was some of the chintzier NWPs. In 3e, it expanded to things like Perform and Profession. And, of course, one of the distinctive features of spellcasting is that two different spellcasters with the exact same basic statistics could be radically different in functionality simply by their spell selection; you could have a sage or a hedge wizard who can't even fight and just throws up his hands if anyone attacks him, or you could have someone who's trying to bend reality and take over the world.
I take it as given that players would choose flexibility over balance every time, if given that choice. The only reason we have any balance at all is that it isn't always a dichotomous choice, and because of legacy elements that still carry forward from the game's wargame heritage.
D&D Next was not designed in a vacuum, and as much as it's moving away from things that some players disliked about 4th Edition, it's also doing a lot more to establish balance and niche protection than they did with 3E.
Yes, and that's a problem. Like I said, you can't unring the bell. Once you tell players that they can make a tauric halfling blink dog or a rogue/warlock hybrid or a noncombatant aristocrat and so on and so forth, I don't think they'll take no for an answer. Mine certainly wouldn't.
Nor should they.
Creating a character exactly to your specifications with point-buy is something you can do in plenty of games, many of which even have the sort of 'classes' as prebuilt packages, but that's always going to come with the tradeoff that balance is never going to be as tight as class-based games. There are just too many combinations of mechanics and circumstances.
Depends on what kind of balance you're talking about. Designing abilities independently from the characters that use them allows them to be very balanced in a broader context.
Even within the extant D&D framework, skills are much more balanced than spells; they have to be. An enormous amount of conceptual space is being funneled into one relatively small piece of skill text, and that skill is going to be usable by most or all characters very frequently, and won't be changed a whole lot by future supplements. So it has to be done right. And most of them were.
Conversely, spells (or, in the broader sense, exception-based mechanical abilities) create new conceptual space with each added spell (exception). They proliferate endlessly and instead of being balanced functionally by examining the practical utility of what they do, they're balanced largely by precedent (the DMG and other sources even say this explicitly), which naturally leads to power creep because precedent is boring, and brings and unreasonable degree of system mastery to the table. In some cases, it leads to blatantly unbalanced abilities that were designed to fill a niche without sufficient regard for their implications in the game world.
It's why 4e, despite having shoehorned characters into such a rigid mechanical framework, is so unbalanced.
But they have to draw a line somewhere, and D&D will probably never draw the line where you want it.
Never say never. It certainly isn't now, but the merits of my thinking exist independently of whatever a company like WotC does. Boundaries, niches, and exceptions are inherently problematic, and consolidated, universal design makes sense. More than anything else, the class-based approach is simply tradition, and like any other sacred cow, I think it's days are numbered. Maybe a large number, but a number nonetheless.