It's one thing for a group to sign a birthday card. A group signing a condolence card? I think that is tacky. First of all, a death requires a letter, not a greeting card. A personal letter, explaining who you are and your connection the the deceased, along with a couple of nice things about him or her-a nice anectdote or a trait you admired are the things the family will take comfort in. There's no comfort to be had with a pre-printed 'sorry for your loss" card with twenty names scribbled on it.
I am going to have to disagree with you. First, if a condolence card was inherently tacky, then everyone I know has been tacky; condolence cards are extremely common and many of them are quite tasteful IMO.
Second, this is exactly what my work department does when someone dies, and this is not a "I do it so it's not rude" justification; they were doing it long before I started there.
Third, it doesn't preclude the anecdotes and other kind words. I have seen many of these cards by my department and they aren't just full of signatures and a printed message, or even just "sorry for your loss"es; they are jam-packed with heartfelt messages. (It was great when a nearby store was selling cards with about 6 pages in them. Finally enough room to not have to write small!)
Fourth, the business setting is not necessarily one where you would have umpteen different explanations of the connection to the deceased. Many would have known him in passing and not really have a story to tell but still feel a sense of loss; even for the ones who knew him, the "connection" is the same for all of them: he was a client of XYZ.
To get back to the OP, I see no problem with what the accidentally left out people are doing.