I think understand what you are saying. It is more of an out put monitor of how much you are "outputting". Like horse power compared to RPM? You can use it to compare yourself to other cyclists, but a HRM would be something that you can use to see how much effort your body is giving.
Well power is the best measure of intensity (or more accurately, power relative to your own capability, such as threshold power, or maximal aerobic power). It's a measure of what we can actually do.
HR has many limitations in that regard (e.g. HR has a max that can be induced despite a rider being no where near their maximal power output). I can make my HR jump 20 beats/min simply by visualising a race scenario while sitting on the couch, yet I am not actually doing anything.
RPM is only one component of power, one can spin fast and not produce much power. The speed of movement has to also be accompanied by force for there to be power. Likewise you can apply a high force (or torque) but if nothing is moving, then no work is done and power is zero.
Power = rotational velocity x torque.
As for comparing to other cyclists, in a general sense yes (e.g. we can compare the power to weight ratio of cyclists, which will give a very good idea of what level of cyclist they are likely to be), but end of the day we have races for that.
I know what you are talking about with your heart rate lagging behind. I will climb a small hill and my HR will not go up much but after a couple of minutes it will start shooting up as my heart try's to supply enough O2 to my body. It will stay there even when I am done with the climb for a bit.
Yes, many of the body's response mechanisms to rapid changes in exercise intensity have a lag time. O2 uptake kinetics, stress hormone responses, cardiac output. Many have a "half life" response time in the order of 30 to 60-seconds.
In incremental exercise tests to exhaustion, maximal HR often occurs shortly after the effort has ceased. Heart is thumping like crazy but you are doing no work.
Then there are other things like the slow component of VO2max which affects cardiac output even when effort is kept fairly steady for longer periods.