I think there are two slightly different cases here. Pre- damaged buds used for budding/ grafting, and on the other hand good buds that get damaged after they have successfully healed and thrown a shoot.
I would use the pre-damaged as a last resort, but if it is a rare scion I bud everything on the stick.
Those good buds that get damaged afterwards will more than likely shoot again, and are worth giving second chances.
The observation by gd banks about the rootstock suckers is probably part of the key. When the buds have fired up they are better primed to keep growing.( and perhaps vice versa )
The practice of bending or breaking the stock to force the growth of the bud is a way to transfer the rootstock energy to our chosen scion, and also used to force lazy buds into action.

In citrus, and many other plants, there are accessory buds on either side of the axillary bud (the dominant bud in the leaf axis). These serve as "spare buds"; in the even something happens to the axillary bud. In some cases the accessory bud(s) will break along with the axillary bud when you force the graft. In that event, I remove the lesser sprout. In most cases, if the accessory bud is lost, one (or both) of the axillary buds will break and become the new shoot.

Ned

Phillip,

In cases where I am concerned about losing the graft entirely, I sometimes supress the less desirable shoot by pinching its tip back, or, if it suits my purposes, I keep it and use it for budwood.

Ned