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.

It wouldn't be too much. It confirms that you are sorry for their loss, and a card can be shared with others in the family who did not hear your words.

Definitely send them a card, and in a month or so, once the novelty has worn off.... you know they're still hurting.
Follow up with them. Ask them how they're handling it. It will mean the world to them.
Remember we are relational beings. our deepest need that can be met by other humans is to be seen, acknowledged, and loved.

Some of the most meaningful interactions are from strangers who took a risk. remember that.

I think it would be very nice.

No it wouldn’t be overkill. You can write your sentiments in the card. Most people appreciate thoughtfulness like that.

If it's a coworker and I just say it and don't bother with a card. If it's a friend or family I will send a card

Certainly not, a card would be very much appreciated

maybe yes