Great thread! Difficult to define terms, though.
Bands where one member who wasn't the lead singer has become very famous for things outside the band, not necessarily musical, and is now the best known person who was ever in the band: Cream, Squeeze, D:ream, Genesis (arguably), not many others.
Bands where one member who wasn't/isn't the lead singer is the best known member of the band because of his/her contribution to the band - I'm not sure I can think of a single one. I don't think it would be fair to say Cream because all three of them made a massive contribution.
There's a Reddit thread about this exact question where someone suggests King Crimson, who I suspect are not that well known now (my husband loves prog rock, I'm a lot less keen but hear quite a bit perforce, sigh). Robert Fripp founded it and is the only member who's always been in it, but he doesn't sing.
Another good suggestion from Reddit: Guitarists Graham Nash of the Hollies and David Crosby of the Byrds. Those two went on to form a little group known as Crosby, Stills, and Nash.