Katia seemed to have very good balance on her blades to help her speed. Pushing/pulling on deep edges is one way to BUILD speed, but another way is to not catch the edge on anything that could slow you down - the latter method was how Katia did it, IMO.
My coach, when he was telling me about pulling the edge, told me that I needed to lean forward more on my forward crossovers. Wait, you say, isn't that how one catches a toepick? Sure, if you lean too much. You'll feel your pick scratch when you hit that point, but right behind the toepick is an area where the blade has the shallowest edges, and when you skate on that area, it'll give you the least resistance on the ice. IMO this was how Katia skated with so much speed and smoothness, but not with a lot of power. Instead of making the speed and working hard to always build it, she was efficient on her feet and kept herself from slowing down.
It's probably not a coincidence that my coach back then used to be a Russian ice dancer.
At any rate, you can either work deep edges all the time a la Patrick Chan to be fast or be efficient and balance on your blades to prevent friction from slowing you down like Katia. Either method works, but you have to do it right each time. Sitting on a deep edge can definitely slow you down!