It was interesting to see how H&H self-opener works so well when opening and closing the action, as compared to just when opening.
I would argue this point. I have yet to see a H&H so called self opener, self open after firing both barrels. The assisted mechanism works when the gun isn't fired but not afterwards. The Purdey system works all the time and once you get the knack of closing it isn't all that difficult.
I'm not sure what self-opening H&H you have inspected but the normal self-opening H&H works totally independently of the fired state of either lock. It is a compressor under the forend which operates at full strength regardless of the state of either lock. I think you are confusing the H&H self-opener with the 'assisted opener' of C&H, normal Boss and Baker '12/20' patent among others.