None of my pets would play with any of the items Major Drill names (they "say" it was too boring). One was an item of normal rarity, two of Common rarity. Is there anything I can do about it? Does it depend on time, are my pets too intelligent? How does it work in general?

That happens when their happiness bar is full (probably because they have a minion or are in the resort). There's not much you can do about it except plan ahead and have one pet out of the resort and without a minion.

: That is good to know, thanks a lot!

You're welcome. :)