First of all, the debate is not about whether high mileage "works." Almost any purpose-driven training will "work." If you were training for 5k and all you did was run 1 mile at 100% effort every day, that would probably work, in the sense that would improve your 5k time if you were dedicated to it. It is only a "stupid" training plan when compared to others that have a higher chance of working. But overall it's a smart plan compared to, say, joigging at 9:00 pace for 1 hr every day.
Now here are the confounding variables in claims that high mileage is the best training plan:
(1) Runners who get injured trying it are not counted in the data. Say we have a control group of 3 low-mileage guys and a control group of 3 high-mileage guys. The 3 low-mileage guys run 14:50, 15:00 and 15:10; and 2 of the high-mileage guys get injured and 1 who doesn't runs 14:30. Then if you only look at averages you would say that the high-mileage guys ran 30 seconds faster.
(2) It is unclear the benefit of easy runs that are longer and easier versus shorter and less easy. For example, a week of training that includes 100 miles at 8:00 pace versus 40 miles at 6:30 pace. Difference in heart rate might be 130 versus 155. Surely there is still a "quality" aspect to easy runs. This variable especially confounding when training for a short distance race such as 5k.
(3) It is easier to reach ideal race weight on high mileage. Ideal race weight can also be achieved on low mileage and calorie restriction.
(4) Cross-training can substitute for real running to some extent. The fact that some pro triathletes run 50 mpw and 28:XX shows that the extent is pretty great. Possibly, the benefits of high mileage are just "high aerobic and vo2 work" which means it's a misnomer.