Good post.
As I read your account, one of the first things that stood out for me was the single ILS receiver. One can do an IFR flight with just one, even without using one at all, but I don't know many professionals who will choose something other than the ILS, given the choice. Two receivers isn't just about redundancy, but about being able to compare one's progress on the approach, one against the other.
Two receivers, however, won't help if one has the same bad signal. I don't have the video presently, but one we've watched numerous times in training has been an account of a south pacific flight that caught a lobe on the ILS, and very nearly flew into the ground. In 2004, a Gulfstream III crashed in Houston while arriving to pick up former President Bush Sr. It was flown by the current Chief Pilot and former Chief Pilot of the company, each high time aviators, each experienced and current in type. Low visibility prevailed as they were turned onto he ILS by vector. They descended into a parking lot, with three dead on board.
I've stopped two pilots in Learjets from following what they thought was an ILS, but instead was a GPS signal, both attempting to descend into mountainous terrain.
It's very important that one observe the altitudes at checkpoints along the approach. Additionally, one should know that based on a generic 3:1 descent into a particular location, where one ought to be. At one mile out, one ought to be about three hundred feet. At ten miles out, one ought to be at 3000'. Five miles, fifteen hundred feet, and so on. If the field is sea level, that's easy. If it's something other than sea level, add those numbers to the field elevation, and there's a rough crosscheck one can do.
Far better than a rough crosscheck is to use the altitudes from step-down fixes along the glideslope, or there fix altitudes, and cross check that one is doing what one's supposed to do. One of our standard call-outs while flying the ILS is to call out the final approach fix "ROBBS checks." The response from each crew member is "Checks." This signifies that the crew member making the call has verified that we're indeed over ROBBS, and that he's verified it by identifying it's position as given on the chart (DME, cross-radial, on the FMS or GPS, etc). He's also verified the correct altitude crossing the fix. The response by each crew member is also "checks."
Our standard calls have long included verifications when passing through a thousand feet. At one thousand above field elevation we called "one thousand," and the airspeed additive ("plus ten," for example, meaning ten knots fast). The flight engineer noted "No flags, descent rate eight hundred." The same calls and checks took place at 500' above the TDZ. The current call-outs only require the calls if something is out of parameters. Otherwise it's only monitored.
Whether one is actually calling out deviations or conditions, or simply noting them, it's important to observe that one is at the appropriate altitude, given one's distance from the runway.
When flying into LAX, a typical approach is ILS 25L, and there are fixes cited on the procedure chart out to 41 DME. This is clearly outside of the ILS limits, but it's very common for crews to pick up the localizer and glideslope out there and follow it down. The FAA published a note a year or two ago about that sort of thing, reminding pilots that the crew still has a responsibility to meet all the minimum crossing altitudes for each published fix along that routing.
When working in mountainous terrain, I usually have a sticky note attached to the panel or control column somewhere that gives me fixes at each mile to the runway, showing what my altitude should be at that point. I can use that as a cross reference for where I ought to be during the descent. I make up the sticky before getting to the terminal area, noting the glide path gradient or the standard.
It was a good lesson for your student, I would imagine, so long as he retained some of what happened during his "frozen" time.
It's interesting to note that you got a low altitude alert by the time you were able to wrest the airplane from the student, disconnect the autopilot, and initiate the go-around. There was a discussion on this board not long ago about touching the runway during a go-around off an ILS, which can and does happen in large transport category aircraft. Perhaps you can see why.