Can you use a magic wand or staff with your Haste action?

Question

Last night was our first use of the haste spell in 5e. One turn, the bard cast shatter with a second-level spell slot, then used his haste action to use a magic staff which cast thunderwave.

I'm not sure if this violates the "only one non-cantrip spell per turn" rule. On the one hand, the haste action says you can "use an object", and a staff is an object. On the other hand, thunderwave is a non-cantrip spell.

The staff in question is a custom magic item, but in this question I'm more interested in the Rules As Written interaction between items-that-let-you-cast-spells and haste "Use An Object" actions. Let's use the "Staff Of Thunder And Lightning" as an analog:

You can use an action to cause the staff to issue a deafening thunderclap, audible out to 600 feet. Each creature within 60 feet of you (not including you) must make a DC 17 Constitution saving throw. On a failed save, a creature takes 2d6 thunder damage and becomes Deafened for 1 minute. On a successful save, a creature takes half damage and isn't Deafened.

Answer 1

No

Haste says:

That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.

Casting a spell or using a magic item fall outside of those options. The rules on activating magic items say the following:

If an item requires an action to activate, that action isn’t a function of the Use an Item action, so a feature such as the rogue’s Fast Hands can’t be used to activate the item.

Additionally, it seems you have a misconception about the spellcasting rules. The rule that limits you from casting other spells in a turn only applies when casting spells with a bonus action casting time. For example, an Eldritch Knight could use Action Surge to cast fireball and magic missile in the same turn.

A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven’t already taken a bonus action this turn. You can’t cast another spell during the same turn, except for a cantrip with a casting time of 1 action.