This is an interesting topic.
My sense is - and this assumes touchpoints (saddle bar pedals) all identically positioned
- smaller frame -> quicker handling (more flickable)
- steeper head tube -> quicker turning (probably because, all else equal, means less trail and less flop, and more forward weight distribution)
- shorter wheelbase -> quicker handling (the extreme cases being a unicycle and a tandem)
- steeper seat tube -> quicker handling (typically means your saddle is further forward in the wheelbase, hence more forward weight distribution, and also can allow a shorter wheelbase)
When you start changing the touchpoints, I think doing things to move your center of gravity forward (longer stem, lower stem, etc) tends to quicken handling.
I also think lowering the saddle can result in quicker turning (your center of gravity is lower, you can lean the bike over more quickly) but correct pedal to saddle distance for your body matters the most.
Overly quick handling isn't necessarily a good thing, but within the range of typical road bikes, I tend to prefer quicker. But if I rode long distances, like touring, I might feel different.
Steeper seattube can allow a small degree of shortening of the chainstays, which in itself shifts weight to the rear wheel, making the steering lighter and quicker.
Moving the rider's body forward has the opposite effect, by putting more weight on the front tire. Charging downhill at high speed, the bike will be more stable with the rider positioned further forward relative to the bottom bracket.
Using a longer stem makes the steering slower, more "solid" and requiring more effort to negotiate tight corners, but it is hard to quantify how much of this is due to the "tiller" effect (described in the next pagraph) of the longer stem versus the added weight on the front wheel.
The center of mass of the rider's upper body can be considered to pivot about the saddle, and with hand contact points being some distance ahead of the steering axis, the rider's mass acts to effect counter-steer and inward-steer in response to each other. So a longer stem would seem to blunt steering inputs by applying some of the pivoted mass's inertia so as to oppose changes of steering direction (either inward or outward).
If the rider's hands were behind the steering axis, as they are on upright motorcycles, counter-steering becomes automatic as the rider simply "throws" the bike to either side without having to counter-steer.