We'll address this in general, by means of a thought experiment:
Alphaworld: A Land Untrodden
Alphaworld is a world in which all magic is rare and powerful (obviously it can't be both rare and weak or the world wouldn't be high fantasy any more). As a result there are no +1 swords, no potion of mildly effective glass cleaning, because minor magic items can't be worth crafting. Anything worth doing by magic must be impossible or at least very difficult to do otherwise, or you would just do it the other, less expensive, way. It's possible apprentice magic users might make a minor magic item or two as they learn the craft, but there can't be too many of them and they can't make that many items or the items wouldn't be rare anymore. In order for magic to be rare as well, Magic Users must also be rare, and they get capital letters in their class title as a result. Most people probably never see any magic during their lives, and the only magic they know of is from legends and stories. Magic items are the sort of thing kingdoms are founded on, and magic is not very well understood. Magic Users are people of renown, prestige, and immense importance simply because of their profession. Because magic plays such a large part in Alphaworld's cosmology, yet is largely untouchable and incomprehensible, Alphaworld has an air of exploration, wildness, and primal mystery. Le Morte d'Arthur and the Dominions video game series have settings like this. Magic Users are not appropriate character choices for low-powered campaigns set in worlds like this, and every magic item should be treated socially at least like a minor artifact is treated in standard 3.5.
Betaworld: A Triumph of Cottage Industry
Betaworld is a land where magic items are exceedingly rare or weak, and may not even exist, but magic is otherwise fairly common. Unlike Alphaworld, magic items don't have to be particularly powerful to achieve a High Fantasy setting in Betaworld, since the ubiquity of magic not divorced from its creator shapes the world sufficiently for that purpose. In Betaworld, mages are an important part of society, but individual mages aren't by simple virtue of their profession the kind of legendary figures mages in Alphaworld were. Because magic is an inherently personal, skill based task in Betaworld (i.e. it cannot be automated) the Industrial Revolution (i.e. the movement away from cottage industry to mass production/standardization) will never happen to Betaworld's magicians. Magicians on Betaworld are craftsmen and artisans, persons who work with style and skill, and the work of an Archmage is fundamentally better than the work of any number of Apprentices and not imitable by a practitioner of less skill. This is a world where the nature of magic itself rejects the quotidian laborer, the Factory, and the Industrial Oligarchy in favor of the skilled craftsman, the Guild, and the Jeffersonian Democratic Republic. Discworld, especially the stories involving the witches, has a setting like this, as does Pokemon (ignoring the Pokeballs themselves). Players can play magic users in Betaworld campaigns, and it is possible that these magic users are not inherently more powerful or respected than non-magic user players.
Gammaworld : The Megacorp
Gammaworld is a world where magic items are common, but other kinds of magic aren't and aren't very powerful. Gammaworld is very much the opposite of Betaworld, and magic is likely to all be Mass Produced via Assembly Lines with Replaceable Parts. Practically everyone is a mage in Gammaworld, simply by virtue of owning magic items, but not all of these items matter very much at all. Nothing magical is unique or special, just serviceable and functional. You can do a lot in Gammaworld, if you have the cash, but it's all easily replicated. Magicians struggle to be different and not just another faceless and nameless drop in the human ocean of people who own all the same items. I haven't seen a setting like this yet. Players playing non-item based magic users are likely to be very underpowered compared to the rest of the party, who should be using magic items.
Deltaworld : A Land of Giants
Deltaworld is a world where all magic is common, but not necessarily powerful. Practically everyone is a mage in Deltaworld, either by virtue of using magic items or by virtue of their own skills and abilities. True mages have certain advantages over those who just make use of magic items, but not so much that the Factory isn't a viable production model. Magic is an inherent part of almost everything that is done in Deltaworld and the players need to be very familiar with the magic system in order to play the game well and avoid confusion. The power level of such a setting precludes it really being medieval, though it may still have some medieval trappings. The Chronicles of Amber (except the part of the first book on Shadow Earth) take place in such a setting, as does every edition of Magic the Gathering. Players not using magic are doing it wrong.
So what changed between the world? First of all, any time one half of the magic system was made rare/weak, the other side had to increase in number/power in order to maintain the High Fantasy genre and the high magic requirement. So making magic items less good makes mages more good, and vice versa. Furthermore, if something is rare, it is unlikely to have much of an effect on the world unless it is also powerful. Worlds that focus on magic items are more democratic (i.e. people are of much more equal value, everyone can make an impact on the world, but one person's impact is rarely world-changing) and industrial (i.e. masses of unskilled laborers use devices more valuable than they are to produce the world's goods) while worlds that focus on magery are more theocratic (i.e. the world is shaped by unique persons of great power) and artisan (i.e skilled craftsmen with years of training produce the world's goods). Also, the more magic is available and controllable overall, the less mystical and more scientific it is.
With regards to the 3.X systems, they generally code for a Deltaworld-like scenario. Removing magic items entirely will lead the system towards Beta world. Removing the purchasing of magic items but allowing their creation through crafting feats and such will lead to an Alphaworld scenario (since you will have to make NPC mages nonexistent or the players would just buy their items from them), except with some of your players playing mages and no reason for the world not to be Deltaworld-like (and your players will probably move things in that direction rapidly). The first of these options would be fine, if spellcasting wasn't so much more powerful than non-spellcasting abilities anyways. If you nerf the magic system as well (maybe requiring two levels per level in a primary spellcasting class) it might be alright. The latter option is just a bad idea. If you are just limiting the availability of some subset of magic items, that should work fine even if you let casters craft them provided the item set isn't something that would make the default power gap between classes even worse, like 'all magic arms and armor'.
As for what games benefit from magic item limitation or availability, games benefit from their setting matching the setting you want them to have. If you want the world to be Alphaworld-like unlimited magic items is a bad idea. If you want the world to be Deltaworld-like limited magic items is a bad idea. Basically, once you understand the consequences of the worldbuilding choices you are making (and whether or not magic items are common is a worldbuilding choice), you will also know what decision to make, based off of what you want in your game. In other words, this part of the question should answer itself.
Definition of Terms:
Any magic that is divorced from a person's conscious work. A stream with magic healing properties, a -2 cursed vorpal dancing flaming falchion, a potion of eased childbirth, and a sentient wax golem are all magic items for the purpose of this answer. An easy guideline to see if something probably counts as an item or not is to think about whether it stops working when its creator dies. If it doesn't, it's probably an item. This doesn't always work, but can be helpful.
Mage or Magic User:
These terms refer to anyone who uses magic, whether that magic is from items or otherwise.