You can apply class abilities to spell casting from magical items
While casting from an item is not the same as a Casting a Spell action, it is still casting a spell. Unless a term is given a special definition in 5e game terms, you use the common definition for that term. You are casting the spell, the ability to do so comes from the item. For instance, the Staff of Fireballs:
.. you can use an action to ... cast one of the following Spells ...
Is no different, in this regard, to Wand of Fireballs from:
you can use an action to ... cast the Fireball spell ...
Jeremy Crawford has posted a number of answers we can point to in the time since this question appeared that verifies this really is the case. Each time he's been asked about a specific class ability the answer has been an affirmative. These were made when his tweets were official rulings, but since have been "downgraded" by the Sage Advice Compendium to his personal advice and perspective. While this is true, it still is clarification of the intent of the designers.
Metamagic
Can Metamagic be used on magic item casting a spell? Yes.
If a magic item's description says you cast a spell from it, you can use Metamagic on the spell. #DnD
and
Metamagic works w/ any spells that sorcerers cast. Wild Magic Surge can work w/ any sorcerer spell they cast. #DnD
Wild Magic
Can using a magic item cause a Wild Magic Surge? Yes
Yes.
Arcane Ward
Can casting from a wand/staff recharge Arcane Ward? Yes
Arcane Ward/Twinned Spell works when you cast a qualifying spell. It even works when an item says you cast one. #DnD
Straight from the DMG Errata:
Attunement (p. 136)
The first paragraph ends with a new sentence:
“If the prerequisite is to be a spellcaster, a creature qualifies
if it can cast at least one spell using its traits or features, not
using a magic item or the like.”
emphasis added
Therefore, for the purpose of attunement:
If a creature is able to cast a spell by virtue of their own ability (not from an item), whether that be race, class, or feature, you are a spellcaster.
Best Answer
This absolutely looks like designers' intent.
The Wands have a variety of restrictions on attunement and they seem to be connected to the rarity / power of the wand.
^ attunement
^^ attunement by spellcaster
More powerful spell wands have a more restrictive attunement requirement. This seems intentional. If you look at the various magical Staves many of them have even more specific requirements than the wands. Like the Staff of Healing requiring attunement by a Bard, Cleric or Druid.
From the rules on creating your own magic item, here is the excerpt on Attunement (DMG 285).
That last point is really important here.
Do I care if the Rogue has 5 Wands of Detect Magic? Probably not. It's not a really powerful spell. Do I care if the Wizard is walking around with 5 Wands of Fireball? Absolutely! What about the Barbaian? Even more so!
The restrictions not only seem intentional, they seem rational and balanced. Especially given how many classes get access to some magic at some point. I think everyone but the Barbarian has some magic using archetype.