If a coworker mentioned a death in their family and you verbally gave your condolences, would it be "overkill" to give them a card?

depends on how well you know eachother and what the customs are.
But a lot of people just do it out here as a curtesy, they ussually send it to the mourning adress or undertaker that is organising the service. If people go to the actual service, they can leave it there in a little basket set up just for cards.