Jimbo, that's a good explanation of the tradeoffs between a more integrated approach and a more modular ad-hoc one.
If you want to build a robot and you know that from the outset, and if you want DC drive motors, for example, an integrated Arduino-compatible board like this is a good solution: http://goo.gl/nksiG
Russell Cameron has designed many Robotic products and brought together what he saw as the most important capabilities for a small robot: FET Motor Drivers, IR receiver, accelerometer etc. See the details here: http://goo.gl/vsqXr
The other end of the spectrum is the "Box of Bits" you mention. One approach is this: http://goo.gl/WTny5
In a classroom environment I have found that 8 students with breadboards means half of the class time is spent fixing connection problems. That creates frustration all around and makes the whole experience much less positive.
The "Box of Bits" of "Electronic Bricks" is used by several Universities and schools. Later when individual projects are done by students, they introduce the breadboard and "Little bits" that need proper connections, pullup resistors etc.
Students see lots of fully-integrated designs: Their cellphones, laptops, video games, toasters, car dashboards. If they are to become designers they need to see the Bits and learn how to bring them together to make systems.
DISCLAIMER: Mentioned stuff from my own shop...