Hi, my opinion, if it is of any help is that you will be doing a lot of work for little real benefit. I like the idea but in practice you will have to put the stored stuff on the new door ( so itself will need to be structurally sound to take the added weight) SO.. When you are down there in the gym , den etc., where is the stored stuff going to be stored. Your wife will get very upset if in the hallway. Also , and more importantly, there will be no safe barrier stopping people falling down the hole when you are down there.
The alternative is to alter the lower stairs for a spiral.. Or ad some winders. That way you still have stairs and may be able to create some space in the small section of wasted space under the existing stairs.
Just my thoughts.