To answer the initial question: yes, some characters do need magic items.
In D&D, there are a number of ways to construct an effective fighter type character.
1. An archer--high strength and dex. Point blank shot, rapid shot, precise shot, far shot, weapon focus, weapon specialization, improved critical. This build focusses on a very high damage output but is generally vulnerable in melee where most of its feats become useless. In addition, the usual lack of heavy armor and the necessary lack of a shield keep its armor class very low.
2. The Spring Attacker--good strength and high dex. Dodge, Mobility, Spring Attack, Weapon focus and specialization with a reach weapon, Expertise, Combat Reflexes, Improved Trip, Knockdown. This build focusses on taking advantage of attacks of opportunity and preventing enemies from taking full attack routines by springing in and out of their reach. Its greatest weakness is that the mobility it favors make it difficult for a spring attack fighter to defend wizards, clerics, or other second rank characters. It is also weak if cornered since the lighter armor and usual lack of a shield keep its AC low.
3. The Damage machine--high strength, weapon focus and specialization with a two handed weapon (usually greataxe or greatsword, power attack, cleave, great cleave, and usually barbarian levels. Close Quarters Fighting is also recommended. This build focusses on dealing out as much damage as possible as quickly as possible. Its AC is often poor but it usually dishes out enough damage to fell any opponent in two or three rounds at most. That's usually quick enough that its opponents don't kill it. Its primary weakness is a party without high level curative magic and monsters that have rend or improved grab or whose damage reduction prevent it from dishing out damage quickly.
4. The defensive fighter. With a weapon and shield, this fighter may focus on a balance between offense and defense or may opt for all out defense. The difference is usually in the magic items since fighters have few non-magical ways to increase AC and have enough feats to have dodge, expertise, and several offensive chains.
5. The dual wielder. This is usually either a variant of the damage machine (longsword/shortsword, twin shortswords, or double sword), or the defensive fighter (bastard sword and shield+shield expert feat. It often adds damage with sneak attacks from rogue levels as well. Its weaknesses include the necessity to tumble about to get flanks which interferes with protecting second rank characters.
6. The Shield Wizard--This munchkinny variant (any of the above) takes one level of wizard or sorceror for the toad familiar (extra hit points) and the ability to use the shield spell (light armored fighters) or a wand/scrolls of shield (full plate armored fighters).
One thing you'll notice about this lineup: the only characters who are set up to actually defend the other party members are the Damage Machine (two handed and dual wield versions), the Shield Wizard, and the Defensive fighter. In my experience, about 1/3 of the fighter types I've adventured with are Damage Machines and 1/3 of them are Defensive fighters. Most barbarians and rangers try to be damage machines and most paladins are defensive fighters. Fighters are split pretty evenly between the two.
Why the discourse? Eliminating most magic items will make defensive fighters an impossible build by low to mid levels (4 to 6). By then, most foes will have attack bonusses ranging from +9 (Monster Manual trolls) to +15 (raging barbarians with weapon focus and a masterwork weapon). An armor class of 22 (fullplate +dex 12+large shield+dodge feat) is barely sufficient to keep enemies from power attacking at that level. It's certainly not enough to keep characters alive long enough to defeat their damage maximized foes. Now the normal defensive fighter of these levels should have a +1 shield and possibly +1 fullplate (for AC 24) and by level 6 will often have an amulet of natural armor and possibly a +2 shield for an AC of 26. That's a high enough AC for expertise to be useful and for foes to have trouble hitting. Without the magic, it's just not possible to boost AC to the point that a character can mount an effective defense.
So, the net effect of scarce magic items for fighter types will be to eliminate the defensive fighter archetype and to encourage the Shield Wizard (the only way to get a reasonable AC).
At high levels, fighters and rogues need armor even more and definitely need magic weapons to overcome DR.
Even more significantly, all characters (except maybe paladins and monks) need saving throw boosts. After all, the minimum DC for a 6th level spell is 19. DCs of 22 (high int) to 26 (high int and greater spell focus) for spells are very common. Since even characters' strong saves are rarely more than +9 without magic (or multiclassing) by 10th level, characters who hope to contend with creatures like Umber hulks (save every round or be confused), mummies, wraiths and shadow mastiffs need to have items that boost their saves if they're going to live through several encounters.
Furthermore, by high levels, all characters need to be able to cope with a multitude of situations. They should expect their opponents to plane shift, polymorph, use improved invisibility, fly and haste, have iron body, spell resistance, deal 10d10 fire damage to anything that touches them, and explode for 10d6 damage when killed. Wizards and clerics have the spells to compensate for this somewhat (and they can make their own darn magic items if they really need them). Fighters and rogues need to rely on magic items for those abilities. If they don't have them, the first sorceror they run into with fly and protection from arrows will slaughter them.