I would trim it off because it will take some effort to keep the broken portion alive and I don't think the resulting esthetic would be at all appealing if successful.
Despite the fact that you've pretty much said all that needs be said about it, Ed, I am in a pedandic mood right now. The following is your punishment for trying to be serious, reasonable, and polite
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Auxin exuded by the buds is the cellular signal in the plant that the branch is still alive. Regardless of the specie, a deciduous branch will die back to the most distal bud (i.e., one farthest from the roots on that shoot). So the shoot will be fine with just that 'new bud below that is set for next year', unless the bud is killed by freezing or etc. (in which case die-back will progress down the shoot to the next bud).
On the other hand, some species such as hornbeams can transport auxin upward in the phloem to keep the only downward, cell to cell, life signalling, Polar Auxin Transport flow going in the portion of the twig below a break but above the most distal bud. These are specie you probably think to not be susceptible to die-back.