Been away for a couple of weeks so just catching up and this interesting thread caught my eye. I agree with the point made by several here, that the public perception of these artists is not necessarily aligned with the reality of their behaviors in life. I believe there is more than a little truth to the persona alluded to by Widespread Panic in their "Tortured Artist" song.
More than a few artists seem to tap into a level of anguish and suffering as a source of creative inspiration and therefore, whether as the result or the cause, artistic people may be predisposed to greater emotional ups and downs in their personal lives than your average accountant or engineer. There are some very cool artists listed here but also quite a few that suffered serious addictions, relationship problems, and depression.
It is interesting how we individually perceive “COOL”. For example, Johnny Cash (saw him in ’75) would be high on my list of “cool” even though he certainly had his personal problems with addiction, but he was faithful to his family and friends, and gave back through his advocacy for prison reform and treatment of inmates, including his concerts at Folsom and San Quentin prisons. I think there is something to the concept that genuine, authentic persons have a cool factor, in spite of their personal failures to cope.
Most of the folks I immediately thought of as having the COOL factor have already been mentioned, such as: Johnny Cash, Roy Orbison, Susan Tedeschi, Derek Trucks, Greg and Duane Allman, Marcus King, Carlos Santana, Boz Scaggs, Zappa, Jimi Hendrix, Jerry Garcia, Bonnie Raitt, Janis Joplin, and so many others.
A couple I did not see here (but may have missed) would include: B B King, Keith Richards, John Popper, Dr. John, and Bob Marley.
Finally, I support @thecarpathian on the inclusion of The Monkees, and especially later on as they grew as musicians, song writers, and bandmates. IMO, while their astonishing level of achievement (the only act to have four No. 1 albums on the Billboard 200 in a calendar year) eclipsed their overall talent, they certainly had moments of brilliance and a “cool factor”. Mickey Dolenz was apparently the first to use a Moog synthesizer on a rock/pop recording, when playing it on Michael Nesmith’s song Daily Nightly from their Pisces, Aquarius, Capricorn & Jones, Ltd. album. The song is about the night the Pandora's Box nightclub on Sunset Strip burned down.