You reckon you can do temperature control in an uncontrolled environment with a handful of transistors? Keep in mind the external temperature is uncontrolled. The yeast itself generates heat within the ferment at varying (and often unpredictable) rates.
The simplest approach you can reasonable consider for the level of control they're looking for in their environment would be a PID controller, which if memory serves me correctly will have more than a handful of transistors in it. You're going to need a pretty decent PID to handle temperature profiling, which their solution
They're using cheap, off-the-shelf parts to solve their problem, which in turn allow them to put extra features in there like web control, and it's now much easier for anyone to do the same thing. The parts may not have existed in 1964, but you need to keep in mind The Doors aren't the only good thing to happen since then.