To rephrase my question in a more specific way; looking at the three dots on Lindbloom's animated 2D-->3D gamut, the upper dot is out of gamut while the lower dot is in gamut (despite the fact that all three dots appear to be in gamut on the 2D view. How would I know which lightness (Y) values would be in gamut for any specified (x,y) chromaticity?

In simplified layman's terms: Colors reproduced using trichromatic color reproduction systems reproduce colors other than the primary colors of that reproduction system by using ratios of the system's three primary colors. The combination of the primary colors in the correct ratio creates the same response in a viewer's eye/brain system as light of the target color would. This works for humans because we have a trichromatic vision system.

Each of the three primary colors has a maximum value. For colors which require one primary color to be brighter than the other two, only the brightest primary color may be at max value. The values of the other two primary colors are limited to values less than their maximum value. Limiting any of the primary colors to less than its max value to hold the correct ratio for a specific color also limits the total potential luminance of the three combined primary colors to less than the maximum potential luminance for the system. Maximum total luminance can only be achieved when all three primary colors are at full luminance (thus producing the perception of white light for the viewer).

If green must be at half the value of red for a specific color, then the total potential luminance for that color will be less than for a color that allows both green and red to be at max value.

Think of it this way. When one is calibrating/profiling a monitor the first step is often to measure the monitor's output when a neutral white signal is sent to it. The monitor's red, green and blue channels are adjusted individually until the measuring device says all three primary colors are equal in brightness. Adjusting one of the colors also affects the relative strength of the other two. If I increase green, for example, the relative portion of the total signal that is red and blue go down. But what happens if I've already got green pushed to 100% and my colorimeter is telling me green is still weaker than red and blue? I can't increase green by any more. It's already turned up as high as it will go! I must turn down both red and blue until they are balanced with green and the output is neutrally white. At that point, that's the brightest the monitor can be and output white light. For any color other than white light, the monitor must be dimmer as at least one of the three primary colors is reduced in strength.

How would I know which lightness (Y) values would be in gamut for any specified (x,y) chromaticity?

The only way you could know is to use a 3D map. It is beyond the capability of a 2D map based strictly on chroma that can be produced by a system at lower brightness levels to show how the gamut for that system decreases as desired luminance increases.

Pure (or almost pure) spectral colours can be generated via a laser (or similar), and appear bright to the average observer. As such, to what do Billmeyer and Saltzmann refer when they suggest that spectral colours must inherently have a very low luminance factor? If this is the case, why is this low luminance (apparently inherent to pure spectral colours, and almost pure colours) capable of appearing subjectively bright?

Billmeyer and Saltzmann are speaking about the ability of trichromatic color reproduction systems capable of producing white light, not single wavelength lasers. Given enough power, it is possible for a trichromatic color reproduction system to produce green light as bright as your green laser. But such a system will necessarily be able to produce even brighter white light when all three primary colors are outputting at maximum brightness. "Low" and "high" are always relative to one another.