I'm very interested in this subject. I'm a regular hillwalker and also passionate about indoor climbing, so I use the muscles in my upper thighs and gluteus maximus a lot! But I often notice when I'm walking up a steep hill that I seem to be engaging my core muscles too. I wonder if that's one reason why losing belly fat helps you increase your pace on the hills too (apart from the obvious reason, that you're carrying less weight).
I lost 31lbs last year after overcoming subclinical hypothyroidism, and my fitness has rocketed since. I have also worked on strengthening my core through yoga, and on strengthening my scapular area, in order to improve my fitness for indoor bouldering.
I notice that friends who are carrying a lot of belly fat struggle to keep up with me on the hills, despite doing as much exercise as I do (or almost as much). I know how they feel, as I was like that before I lost my hypothyroid weight. No matter how much exercise I did, I would huff and puff my way up the hills, and go pretty slowly. This was partly due to being hypothyroid, but I also feel that the core area is key to this too, linked to belly fat.
I often skip up the hills now, and it feels as if the power is coming from my core - or as you say, from my leg muscles acting through my core. It feels to me as if the core is key to good health and fitness. I would love to learn more about it!