When you cast a spell with the Spellbind trait, you may immediately bind a spell to it.
You are not required to bind a spell when you cast an object with the Spellbind trait. It is optional. Thus, the recipient of the Elemental Wand may leave the wand without an attached spell when they receive it.
However, I am not able to locate any rules regarding unbinding a bound spell without replacement. Without such a ruling, I would say that neither the Tower nor the Wand could remove a bound spell without replacement after their initial cast. Both cards have the wording that you may "change" the bound spell, which I would interpret as requiring a replacement, but I do think there is some ambiguity there that could use an official ruling.
Wildhorn is incorrect about the Tower. Even though it is a Familiar, it does not prepare it's spells like other familiars do. Therefore the spell does not return to it's owner's spellbook during the planning stage because it is not a prepared spell.