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.