Syd Barrett, being the principal songwriter, the guy who came up with the band's image, the band's sound and even came up with the name "Pink Floyd" was absolutely THE leader of the band. Perhaps he always had problems underneath, but the psychedelic drugs he did certainly helped his creativity and exasperated his mental problems. It is really remarkable that the band even survived Barrett's breakdown at all and made it out of the late 60s London underground music scene. They depended upon him so much, but Roger Waters stepped up and took over the band - for better or for worse, depending on whom you talk to!
Carrying over to the Beatles, this proposed breakdown would have to happen just as the band was getting big. If John Lennon went off the deep end in lets say, 1968, I'm sure the band would just break up and call it a day. They've done what they've wanted to do beyond their wildest dreams anyway, so why carry on? However, if this breakdown happened in lets say, April of 1964, the Beatles might have to take on a new member to fill in for John. At this point in their career they were just hitting it huge in America and elsewhere, and the "machine" was too big to just stop, and too much was riding on it. David Gilmour slowly began to replace Barrett as a direct result of Syd's increasing erratic performances and attendance to gigs, and the disastrous tour of the U.S. in the spring of '68 eventually replacing him as the guitarist. Another example of this could be seen with Bruce Johnson filling in for Brian Wilson on the road with the Beach Boys after a series of mental breakdowns. Sometimes bands are bigger than one of the members.
Two things could happen here. One, the Beatles could outright kick John out of the band (which I can't see happening), or two, they could retain him as a song writer and have someone fill in for him on the road - leaving John at home with nothing to do but write and deal with his problems the best he can. One thing can be sure, the entire concept for the album Help! would never have been written, since it was all about the pressures of fame, having to live your life in the public eye and their brutal and horrific touring schedule. Paul would then be free to take the mantle as "leader", something he had wanted since they were the Quarrymen and the Silver Beetles anyway. John would be looked at as some reclusive genius and a mysterious artist, which is how Syd Barrett and Brian Wilson are both looked upon today by many people...
Interesting thread topic.