Let it be stated upfront that I am not asking to remove the 24-hour waiting period for reward-type bounties. I am not asking for the ability to pre-select the winning answer, either. My request is this:
When a user sets a bounty to reward an existing answer, existing answers should be eligible for auto-award of the bounty.
It makes no sense to deny eligibility to the answers that the user explicitly wanted to reward.
One example: Math.SE currently has three 500-point bounties set by a departed user, with the stated reason "reward existing answer". They are set to questions with outstanding and very thorough answers; however, those answers will not receive bounty now that the originating account has been deleted. (The bounties are still there, owned by Community and subject to auto-award rules.)