No, there are not more than enough items if you want to give out items of the exact levels indicated in the DMG for characters above first level.
Checking the relative levels of magic items I have to agree with you. There needs to be at least one uncommon item allowed.
The important thing is to make sure that armour/neck/weapon implements do always fit on the +1, 0, -1 progression.
Lets check it for 2-5 for basic character types:
Light armour (1,2,3),weapon (1,2), neck, (1,3,4,7):
Level 2: Sylvan armour (3), Defensive/Vicious weapon (2), Amulet Protection (1). There exists also (1), (2), (3) and (2), (1), (3) options for this build.
Level 3: Sylvan armour (3), Defensive/Vicious weapon (2), collar of recovery (4)
Level 4,5,6: Lacks level 5 item (but can buy the missing armour/neck/weapon with gold)
Heavy armour (1,2,4),weapon (1,2),neck(1,3,4): (armors are
Level 2: Easy. You have options of doing (1),(2),(3), (2),(1),(3), (3),(2),(1) and (3),(1),(2)
Level 3: Black Iron armour (4),Defensive/Vicious weapon (2), safewing amulet (3). Note: Doesn't work for chain users.
Level 4,5,6: lacks level 5 item (but can buy the missing armour/neck/weapon with gold)
Implement users on other hand will be safted, if they don't use staff, rod, totem or holy symbol.
Light armour (1,2,3), implement (1), neck, (1,3,4)
Level 2: Veterans armour (2), magic implement (1), Safewing amulet(3). Gets more options if uses staff, rod or totem
Level 3: Not doable unless uses staff, rod, totem or holy symbol
Level 4,5,6: lacks level 5 item (but can buy the missing armour/neck/implement with gold) Exception: For Totem users these levels work too.
Of the non-basic magic items, only the Safewing Amulet is usable by all characters. That item covers third level and every five levels thereafter.
Some characters may have options at 2nd, 4th, or 5th levels, but many will not. A player character starting out at 4th level may very well have the Safewing Amulet as the only level appropriate common magic item that he can use.
I assume when you say usable, you mean useful. Levels 4,5 and 6 are the problems with common items as there are no common level 5 armor/weapon/neck slot items. And only level 5 common implement is a totem.
I have to check the rest of the common item list to see how many level 3-5 items there are and how many of them are usable for all builds to do proper analysis.
But after doing quick check I have to agree, DM needs to give out one uncommon item at least. Or we need to get level 2-5 common items for all required magic items. Preferably both.