The original film the Leopolds invented was a two-color system (two-color schemes had a long and varied history, starting prior to the Leopolds' work, and continuing long after; even in the '50s or '60s Polaroid did two-color experiments before finally coming out with three-color Polacolor).
By the time Kodak released it, it was a three-color system, but the process bore little resemblance to everything used since the short lifespan of that initial version.
The processing consisted of successive bleach/develop/dry cycles, with each bleach stage being precisely timed. The stories of the Leopolds using a metronome to time the process most likely had to do with the extremely critical bleach stage timing.
For the first stage, aimed at processing the deepest layer (the layer next to the film base), the film -- all three layers -- was bleached, re-exposed to white light, and then developed with a developer that contained a silver developer, a color developing agent, and a color coupler for that layer's intended color (Yellow, I believe).
After the development (of that stage) was complete all three layers were processed -- and, they were all colored with the same color dye!
The film was then dried (after a wash, and possibly a stop bath prior to the wash).
After it was dry, it was given a very precisely timed bleach bath. The timing was critical, because the bleach would not only rehalogenate the silver image, but it would also destroy the dye image (shades of Cibachrome!)
After the exactly right amount of bleaching, the top two color layers (Cyan and Magenta, IIRC) were bleached, leaving ONLY the bottom layer developed and dye-colored.
The process was then repeated (this time, the top two layers were developed and dyed the same color (Magenta, I believe).
The third (final) step required a shorter bleach time, since only the topmost layer was to be bleached and redeveloped.
The whole thing sounds a bit Rube Goldberg, and as you can imagine everything was extremely critical. If anything was the least bit "off" (time, temperature, agitation), the critical bleaching steps would either penetrate the film too deeply, and affect the already-completed layer(s) beneath, or, not penetrate deeply enough. Either case would wreak havoc with color balance, density, etc.
This process was what comprised Kodachrome from the time it was released in the late 1930s, until the second incarnation which followed a few short years later, which used the selective-reexposure system that is still in use today. Gone was the need to repeatedly bleach and dry the film. In the place of that incredibly complex and demanding system, a fairly "simple" system with much higher reliability and color purity was released.
When you look at very old Kodachromes, you'll find that some of them look pretty awful, but others, just a year or two less-ancient, will look like they were shot yesterday. That's the cutoff point between the two processing systems.