This is simply not true. That additional power will never be used unless you increase the volume. Extra power only does one thing--increase the volume. There is no magic pathway for power to follow that increases sound quality without increasing volume.
The above statement assumes that an amp is operating within its current and voltage limitations. There could be cases where an amp cannot provide enough current or voltage into load A even though a calculation based on power into load B suggests otherwise . But that is just a matter of using the correct calculation and enough data--you may need more information than only an amplifier's published specs.
Please note that I'm not saying all amplifiers with similar power specs sound the same. I agree that some headphones "scale" with amplification. But it's not a question of power.
Also, nice work OP. This is a great resource as long as it's not taken as the final word in whether or not a given amp will sound good with a given headphone.
You might want to read up on "headroom" in regards to your first paragraph. Just Sayn'