I’ve long wondered when a famous band like The Rolling Stones would become a brand with one or more touring acts of non-members. Now I think these guys have out-lived that opportunity. My kids like songs from this era, but I don’t think they feel a connection to any of the bands themselves.
Maybe the bands will live on as aesthetics – Beatles patio furniture, Rolling Stones hot wings, Led Zeppelin phone cases. We see a bit of this now with all of the beers branded with hard rock acts.
Back far enough, symphony orchestras and chamber groups. The original Phil Harmonic must have died years ago.
The original (pre-Justin Hayward) Moody Blues (“Go Now”) line up has all croaked.
The Kinks were a four man group. Was one of them involved in a bizarre buzz saw incident?
Two-thirds, if counting by surname, is accurate. 
The Four Tops. All gone.
How many people in this thread looked at Mickey Dolenz’ wiki before posting?
Original bass player was Kelly Garni who’s still alive.
Depending on your definition of “big rock band”, Taste (Rory Gallagher, Eric Kitteringham and Norman Damery)
With David Crosby dead, I thought maybe The Byrds, but both Roger McGuinn and Chris Hillman are still alive in their 80s.
And, of course, Stills, Nash, and Young are all still alive.
I guess lots of money helps with longevity. Still pretty amazing.
The Spinners are still touring and I don’t think they have any living original members.
A singer named George Dixon replaced C. P. Spencer before they renamed themselves the Spinners from the Domingoes in 1961. I can’t find any info on whether he’s dead or alive.
All the named band members from the first album are dead but Al Kooper is alive. He produced the album and played on 5 of the 8 songs. That’s him playing organ at the beginning of Free Bird.
Molly Hatchet had 6 original members during their 70s heyday years, and all 6 are dead now. There is a band called Molly Hatchet touring now, but there is little connection to what I think of as the band.
Motörhead
This won’t qualify as “biggest,” but they are noteworthy — one of the (if not the) prototypical surf rock band, The Ventures. They’re being kept alive by a guy who joined in 1980 but the original 1958 quartet is gone.
Pretty sure all the original members of The Mothers of Invention are dead.
Looks like The Moody Blues qualify - Justin Hayward is still alive, but he only joined in 1966 when Denny Laine left the band.
The current iteration of Foreigner has none of its original members. Ed Gagliardi and Ian McDonald are deceased, and the others have either long ago left the music industry, or in the cases of Lou Gramm and Mick Jones, who IMNSHO were the nucleus of Foreigner, can’t tour or perform due to health issues.
ETA: I see someone already addressed that. Since the 1990s, Gramm has been dealing with the aftereffects of a brain tumor that he was literally born with (ice skater Scott Hamilton had the same thing) and removing it destroyed his pituitary gland.
There are lots of bands with only one surviving original member, the Grateful Dead most recently earning that distinction, but finding a band that actually meets the OPs criteria is more difficult. After an hour of googling, I give up. 