The primary reason you see the 6 min effort cited as the pace for VO2max is because, for an effort the lasts longer than 10 min, the individual will likely not achieve VO2max. If one picks an arbitrary distance (e.g. 5 km), some less elite runners will take > 15 min to complete that effort, and they will likely not reach 100% VO2max, and if so, just near the end. Even most elites, won't spend much if any time at VO2max if running at 5 km pace. If you perform intervals at that pace for much shorter periods, they won't be at VO2max. So, if you're training at 5 km pace, you're training at 5 km pace, but likely not at VO2max.
Why 6 min? It's long enough to allow time for VO2 to evolve to max (requires at least 90 sec), but not so long that you don't get there. BUT, depending on how you test for vVO2max (e.g. Billat's approach or someone elses), it could be 4, 5, 6, even 8 min (tlim @ vVO2max). 6 min is in the sweet spot where you know you got to VO2max, but not so long that the intensity is too low.
The OP refers back to texts such as JD and Pfitz/Coe, who use the 5 km pace, or similarly lenghty efforts to determine VO2max training pace, and this likely comes from the view that the best way to train VO2max is to train slightly under it, as to not incur too much strain. Whether this is the correct approach or not, we will likely never know for certain.
Steve