According to this answer to a question which asks in part for help to "...understand the functional distinction between the 7-day period and the grace period from the user's point of view...":
The principal distinction deals with two things:
- The question stops being advertised as a bounty once the grace period starts. This means that it is now far less likely to get an answer than when it was being advertised.
- If a user posts an answer during the unadvertised grace period, then only the person who set the bounty can cause it to be awarded to them. It is ineligible for the automatic system.
So uses might include additional time for the bounty creator to think before acting, and additional time for people who already know of the bounty's existence to potentially finish and post or modify their answers, knowing(?) that if posting now, they need to convince the person who created the bounty and that the a auto-award system will not consider them.
Point number 2 shows that the bounty auto-award system is blind to new answers posted during this grace period.
So, considering the functional distinction between the original 7 day period of the bounty, and the newer 24 hour grace period, I would answer yes, there is utility.