The various magic rods, wands, staves, etc. in the DMG all have certain abilities. But do they also count as arcane foci for the purpose of not needing material components for all your other spells?
Yes. Magic items in the DMG refer to mundane items listed in the PHB, and use the corresponding stats. Staffs, rods, and wands are listed in the PHB as categories of Arcane Foci, therefore anything listed as one of these types of items in the DMG is referring to an Arcane focus unless otherwise stated.
If you were to assume that items in the DMG do not correspond to the stats listed for them in the PHB unless explicitly stated, you would have to assume that all armor, shields, and weapons in the DMG have ambiguous stats. That would mean that all of these items are not actually armor or weapons unless explicitly stated, which is absurd.
Some items are explicitly listed as arcane foci, such as the Tome of the Stilled Tongue. This is because they are an item type that is not listed in the PHB as an arcane focus.
According to an unofficial tweet by rules designer Jeremy Crawford, you can use a Wand of Magic Missiles as a focus.
Dave Ciskowski: Can a wizard use a magic wand (e.g. wand of Magic Missile) as her arcane focus?
Jeremy Crawford: Yes
The second line of every magic item entry in the Dungeon Master's Guide tells you the kind of item and its rarity. If it's described as "Wand, rare" for example, then you know it can function as a normal (non-magical) wand. Similarly for items designated as "Weapon", "Armor", "Rod", etc. Items that don't have any such function are always designated as "Wondrous item".
The Tome of the Stilled Tongue is described as "Wondrous item, legendary" so its usability as an arcane focus is exceptional and is explicitly called out in the text.
A potential point of confusion is that a "staff" is an arcane implement, whereas a "quarterstaff" is a weapon. According to page 140 of the Dungeon Master's Guide:
Unless a staff's description says otherwise, a staff can be used as a quarterstaff.
...but not vice-versa.
No. An arcane focus is made specifically to be an arcane focus. DMs can houserule this differently, but by default an item must be explicitly constructed to be an arcane focus (PHB, p. 151; emphasis mine):
An arcane focus is a special item — an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item — designed to channel the power of arcane spells.
An object just happening to have the form of an orb, crystal, rod, staff, wand, or "some similar item" (which covers a lot!) isn't enough for it to be used as an arcane focus. Finding a crystal in a cave, a shepherd's rod by the side of a field, a child's glass marble, or a little jade statue of a wizard doesn't mean you've found your new arcane focus.
Similarly, although magic items might happen to take one of these forms, they are not specially constructed to be arcane foci. Yes, they're specially constructed — but they're constructed to be a specific magic item, which has a different magical function from an arcane focus designed to channel outside spells.
(Someone might argue that magic items are already designed to channel some kind of magic, so why not outside spells. To this I answer that arcane magic is arcane and detailed in the particulars of how it works, not wishy-washy — the construction matters very much for what kind of magical use it later has and isn't a general purpose "vessel" for any kind of magic at all. Besides, if being merely enchanted at all makes a wand of fireballs usable as an arcane focus wand, then the broadness of "some similar item" would feasibly make a stone of controlling earth elementals or a wind fan usable as arcane foci.)
Just like normal items, magic items have to explicitly say they may be used as an arcane focus in order to be used as one (unless, like a bard, you have a class feature that says otherwise). An example is the Tome of the Stilled Tongue (DMG, p. 208), which says that it can be used as an arcane focus once attuned. This explicit exception proves the rule that magic items cannot be used as arcane foci unless they are made to be.
Jeremy Crawford, official rules designer for 5e, made a clear (but now unofficial) ruling on this question on Twitter in 2017, in response to a now-deleted tweet:
Can Wand of the War mage be an arcane focus? Doesn't specify, and while wands can be foci, it's not clear that ANY wand can
Any wand can be used as an arcane focus.
No: a magic item has to explicitly specify that it can be used as spellcasting focus.
An item has to be designed to be a spellcasting focus, and its description must state it clearly.
There are several magic items that do not belong to the list of types (orb, wand, crystal, staff, rod) but that can be used as spellcasting focus: for example, the Alchemical Compendium:
[...] While you are holding the book, you can use it as a spellcasting focus for your wizard spells. [...]
But there are magic items whose type is among of the allowed ones for arcane focus. The Astral Shard is a crystal and its description explicitly states that it can be used as a spellcasting focus (emphasis mine):
This crystal is a solidified shard of the Astral Plane, swirling with silver mist. [...] You can use the shard as a spellcasting focus while you hold or wear it.
A similar description is given for the Elemental Essence Shard.
The Orb of Shielding reports (emphasis mine):
An orb of shielding is a polished, spherical chunk of crystal or stone aligned to one of the planes of existence. If you're a spellcaster, you can use this orb as a spellcasting focus.
The Ruby Weave Gem is clearly a crystal, and its description says (emphasis mine):
While you are holding this gem, you can use it as a spellcasting focus for your spells.
SevenSidedDie's answer already cleared this out several years ago, with solid bases in the rules: here more support is provided showing that several magic items are of orbs or crystals and they clearly specify that they can be used as spellcasting foci. To the best of my knowledge, there are no magic wands with such property.
Several answers cite Jeremy Crawfords's tweets: but they are no more considered official, hence are not valid as a sole support of an answer.
The Wand of the War Mage does not specify that it can be used as an arcane focus, yet is clearly designed to be used with the caster's own spells. We can reasonably assume that this device can be used without another arcane focus, despite that it does not specify this fact.
There is no actual rule on the matter, and it should be up to the player and DM to determine how they want to handle this. Ultimately, the DM decides the issue.
The 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.