Would you allow a spellcaster to cast Misty Step and then use a Wand of Lightning Bolts, for example?
No. Wands explicitly state one "casts" the associated spell. This choice of language is purposeful.
The DMG page 141 answers your question. Under
Spells: Some magic items allow the user to cast a spell from the item. The section goes on to detail that magic items like Wands change the rules for how spells are normally cast, bypassing the need for components and using or having a spell slot (and because features like
counterspell require the spellcasting to be perceived,
a wand bypasses this...it has nothing to do with whether a wand is a cast spell...it is).
Further, magical items, including Wands, are not considered "Objects" for purposes of the rogue "fast hands" feature (same page and 2020 D&D official sage advice errata). Some future DM screens would list this as "Use Magical Item" for clarification of rules that already existed.
Note: 5E wands vary quite a bit from how wands worked in prior editions. Previously, they independently triggered a spell for spellcasters and those who had a class feature to bypass that requirement. Now, anyone can "cast" the spell. Sage Advice through Crawford felt Wands should trigger class abilities like
arcane ward but this never made it into official errata, leaving it a DM question. Not your original point, but worth mentioning I felt.