Coming from a sport fencing background, it's been etched in the stone of my mind that footwork (and its sub points like distance and timing) was the all-important factor of sword combat. Are there any sword styles all about handwork or strength, etc?

I can't think of any traditional sword art that doesn't heavily emphasize footwork, timing and distance.

The major difference between classical and sport fencing is the nature of attacks. In sport fencing you just need to tap them in a scoring zone. In traditional fencing you can hit them anywhere, and you must realize that weak, fast strikes probably aren't going to hurt someone. Also, your weapon is going to be vastly heavier making a lot of your foil and epee techniques impossible. Interestingly classical fencing has vastly more slashes than jabs. Wide, circular slashes.