What a great way to have fun, make mistakes and produce a necessary piece of bench equipment. And, you don't need a multi-domain digital scope to do it! Actually, you don't need a scope at all. (Ok, the supply might oscillate and throw you a curve ball, but you can spot the symptoms with a good multimeter.)
I wasn't trying to re-design Dave's circuit, but instead I was speaking in general about getting a single linear pass transistor voltage regulator to operate near 0 volts output. The LT3080 does all this for you, but you haven't really learned much about power supply designs. Yes, you can use your voltage reference or current reference to tighten up the voltage supplied to the adjustment pin on your LT3080. But just for the sake of ground-up design experience, lets assume you are working with all discreet parts.
Any transistor BP or FET will have a very non-linear turn-on characteristic which will require a feedback circuit with enough loop gain to control your device at those very low levels of base current (or gate voltage). Without closed loop operation, you are depending on the transistor's Beta which will swing widely around the turn-on point and with temperature. And what better way to provide the necessary voltage AND current feedback than a current sensing resistor in series with the output? The "improved" LT3080-1" has the sense resistor built right into the package (by popular demand according to the manufacturer).
The usual configuration for the regulator is to have the NPN emitter on the output side of the supply. A power MOSFET would be an even better choice for low voltage output because of the smaller D-S leakage current when the device is near or at turn-off. Just another idea for fun experimentation, not set in stone.
Finally, the use of a current sense resistor in the output is not the horrible idea that most people believe it to be. Taking maximum current output into consideration, the sensing resistor can be (and often is) well under an ohm. 1 ohm was chosen in the video to make calculations simple. A precision length of circuit board trace is just what the doctor ordered.
Dave mentioned an important point to keep in mind - don't get carried away with the size of your output filter capacitors or any capacitors used in the feedback circuitry. That way, the supply's ability to respond to dynamic load variations will be enhanced.
45O