Regarding Medeski, this is the best interview I've found where he talks about why/how he values the pre-midi, electro-mechanical stuff: https://web.archive.org/web/20130106235326/http://www.emusician.com/news/0766/being-john-medeski/146513
Most relevant bits are toward the beginning:
"Medeski's idea of making music better includes surrounding himself with an array of vintage electronic keyboards, effect pedals, and amplifiers. His main instruments are a Hammond B-3 organ, which goes through a Leslie 117; a Clavinet that goes through a wah-wah, a Roland Space Echo, and a 1953 Fender Bassman amp; a Wurlitzer that's sent through a 1957 Fender Tremolux; a Mellotron that goes through a 1968 Fender Pro Silverface; an ARP String Ensemble; a Yamaha CS80 synthesizer; a Melodica, which is another Wurlitzer-type keyboard (“I don't know what the exact name and model number are, but it's somewhere between a Farfisa and an ARP,” says Medeski); and a Steinway piano. His only concession to modern times is the Moog Voyager, which he puts through a 1950s Kay 703 guitar amplifier.
“I'm more interested in an instrument that has one sound world rather than an instrument that has 1,000 sounds, but each type has its limitations,” explains Medeski. “And each of my keyboards has its own sound world. I think of them as different personalities, and that's why they each have their own amplifier. I spend a lot of time finding amps that work. I really like the Wurlitzer sound that comes from its own speaker, but you can't mic it live. Even in the studio it doesn't have enough balls. But in the studio the Wurlitzer and the Mellotron can sound great direct to tape.”"
/end quote
What I find interesting is how specific he is with the amplifier/keyboard, how deliberate the choices seem to be. I think he hardly ever uses a Rhodes, for instance. And he seems to prefer the pre-200 Wurlitzer models. Fascinating!
And I think what everyone is saying here is simultaneously true: there is a degree of fairy dust with these old instruments, no doubt inflated by nostalgia. They are heavy and impractical. There are reasons the technology changed. But playing them is a unique experience, despite the fact that the emulations get better every day. Anyone who gigs these things is probably deluded, and heroic.