Smallboatlover, a boat dancing on the ball is a big deal for two reasons. First it is really unpleasant (I have experienced 120 degree swings heeling over 20 degrees on each side). Second, it can greatly contribute to dynamic forces on your mooring or anchoring gear increasing the chances that it will chafe, break, or drag.
billyruffn, I suspect that the answer to your question lies in geometry. Lines
(and chain) can only apply a force in line
with them. As your bow starts to fall off, some of that force becomes a restoring force to pull the bow back up into the wind and some holds your boat from moving backwards. With a bridle, the proportion of the restoring force is lower at first until the line
goes against the bow and then it is equivalent to having the line
over the bow roller. The worst is when you have it attached to only one side a few feet aft of the bow. Try this sometime and watch the geometry change as you swing one way and the line goes against the bow and the other and it stands straight out. Ideally, you would have the line attached to a point out in space that is far forward of the bow as that would provide the most restoring force. You should get similar results if you don't have a bowsprit if you pull the bridle up tight as described by RichH as with going over your bow roller. The reason that this is not an issue on cats with their bridles is that the angle is so wide on the bridle that the point where it meets is essentially fixed in space, the geometry never changes. I hope that this makes sense.