I see the point in the 48 hour wait. We definitely don't want to see a featured tab that is filled with a series of low quality posts that are unclear, incomplete, or unanswerable simply because of missing details.
One of the things that I enjoy about the bounty questions is that there is generally lots and lots of good details in the questions, and the questions themselves generally produce really great, detailed, quality answers.
In this regard I strongly feel that the system isn't broken and is working as it should. If the threshold were to be lowered to 24 hours, would that lower the quality of these questions or create a series of frivolous bounty questions that really aren't worthy of a bounty? This is a very valid concern and one that should be considered carefully.
It would be a tragedy for the reputation system to suddenly be diluted by users who give normal, one or two sentence, hit and run answer on categories of questions that have traditionally been answered with great care and attention to detail.
How to make this happen without sacrificing quality?
However, StackOverflow is a system that is built on the concept that the more reputation you gain on this platform, the more responsibilities and trust you earn.
Thus, here are a few possible proposed solutions that may mitigate the very valid concerns of lowering the threshold to 24 hours, while still providing the community with the option to post a bounty at the 24 hour mark:
1 - Require that a user have at least 10k reputation (or 3k or 5k) in order to post a bounty prior to 48 hours.
These users know -- or should reasonably know -- what constitutes a good question and will more than likely only post bounties on good questions.
2 - Limit the posted bounty amount to no less than 500 reputation. If the question isn't worth a bounty of 500 reputation, then the asker can wait until the 48 hour mark.
Desperate people should theoretically be more willing to sacrifice more in exchange for their desperation. This will eliminate frivolous bounties posted by users who just need to avoid waiting until the last minute (because of lack of planning or preparation) and reserve this type of bounty for those unique situations where a bounty would really help.
3 - Require that a user convince a 3k or 10k user approve bounties posted prior to the 48 hour mark.
Again, this could help provide the necessary community controls, which work really well for closing and other moderation issues, to effectively prevent abuse of the bounty system. Posting a bounty prior to the 48 hour mark could for instance require a 3 to 5 vote approval by high rep users.
Additionally, another thing that could help ensure these users don't just approve any bounty is to consider removing 2 reputation points -- from the users who approved the bounty -- for every bounty that isn't awarded by the person posting the bounty. In other words, 3k and 10k users will only approve the bounty if the question is clear enough to where the question-asker will likely get a good, solid answer. At the same time, the reputation hit won't be so significant as to be more than a simple downvote.
NOTE: I don't think all three of these need necessarily be implemented in order to make this a success. Any combination of these 3 suggestions could be beneficial and allow for a special 24 hour threshold.