I'm with 'the other school', but your terminology is a bit odd. People training with power meters use functional threshold power (FTP) to define training zones. FTP is sort of your power at lactate threshold - its the power that you can maintain for an hour. I've never heard of anyone training with power using VO2 terminology. The aim of the game whether you're using power or an HRM or just perceived effort though is to increase your FTP (even if you don't actually know what it is) so that you can maintain a higher speed for the same effort.
But the question which you've avoided anyway was how does glancing at an HRM a few times in a ride help you train?
Granted. It depends on what you define as "Effort". Without any changes in mass and aerodynamics, the effort to maintain a desired speed remains the same, in kW terms.
If your HR is less, you are 'fitter', more able to transport O2. Your stroke volume may have increased slightly, but its mainly due to capillarisation getting O2 to the muscles.
To get O2 into the blood, your pulmonary system will have to develop, so as this is where O2 enters your system, it should be the most efficient part of the process and deserves the most attention.
The answer you are most eager to hear is... It doesn't. Knowing your HR when training is neither a here nor there. What you want to know when training is "Am I more powerful than last week? and did it 'feel' easier?"
As has been quite rightly stated, each person's HR Max is a personal thing. Once a person knows theirs, it reduces by 1 BPM each year.
Keeping a track on any increase in your HR max is 'a regular torture session', which should be supervised by a fitness professional. Going to these lengths on a weekly or fortnightly routine is... well, you must be being paid for it.
If cycling is not a person's financial livelihood, a HRM is a nice little extra for making a bike ride a bit more interesting.