Justin, theThe only item in the DMG that specifically says it can be used as a focus is the "Tome of the Stilled Tongue", which says "If you can attune to this item, you can use it as a spellbook and an arcane focus." Tomes aren't normally foci, so that line would be needed.
The Instrument of the Bards entry includes the line "This effect applies whether you are using the instrument as the source of the spell or as a spellcasting focus."
Of course, bards may use any musical instrument as a focus, so that doesn't really help us. But it is the only other item in the entire DMG that mentions being used as a spellcasting focus.
The biggest point in favor of the default Staves/etc being useable as foci is the fact that many of them grant bonuses to spell attacks. Since you need a hand free for somatic components, and may substitute your focus for material components, you can't cast a spell while wielding a staff and separate arcane focus.
So, I'd say that any magic item of the types listed may be used as a focus. Thus, Staff of the Magi yes, Wind Fan no.
And I'm totally fine with the Wand of Fireballs. It's a wand that channels arcane power.
In the end, the rules are unclear, so it's a DM's call. I just think it's silly that a wizard can't cast a spell while holding an item specifically designed to be the best item in the book for arcane casters.