In a low magic campaign, I think the "weak" magic item should be the items that are more rare.
The more powerful ones are seldom in either kind of campaign, but the weaker items are what makes magic commonplace - everyone (except the average commoner, at least) can buy a scroll or a potion, but even in a high magic campaign, a +5 Longsword of Dragonbane is rare, since only so few people can buy it.
If you limit access to the cheaper items, people begin to focus on a few, single items. Instead of owning Boots of Striding and Springing, Belt of Giant Strength +4, a +1 Full Plate, a +1 Shield, a +3 Battleaxe, 6 Potions of Cure Light Wounds, 2 Potions of Cure Serious Wounds and 4 Potions of Bears Endurance, a Fighter might have only a Full Plate +2 and a Longsword +5. These items are still very powerful, but magic is still less visible.
After a magical cataclysmn, "powerful" magic items can still be found (but not made), while weaker items lost all their power (higher level magical items have better saving throws, after all

).
It will probably also be easier to balance encounters (especially with monsters) if characters still have access to usable items according to their wealth, instead of even being underequipped or using a lot of "one-trick" powers (that might either weaken or strengthen the characters in the specific situation beyond anything predictable)