I'm no bounty hunter (or bounty starter), but I agree with you.
- Let's assume the bounty starter is a male so I can use "his" instead of "his/her".
- I use "post-bounty" to mean "after the bounty is started"; "pre-bounty" to mean "before the bounty is started".
First of all, let's consider the possible reasons why a bounty is not awarded within the stipulated time.
- bounty starter doesn't think any post fits his requirement.
- bounty starter forgets about the bounty / doesn't login to SO for that period, etc. (i.e. a reason not related to the answer not fitting his needs)
Currently, if the bounty is not awarded within the stipulated time, the highest voted answer (>2 votes) after bounty is started gets awarded half the amount, whereas the highest voted answer pre-bounty is ignored.
Let's consider reason 1 from above. Both highest voted post-bounty answer and highest voted pre-bounty answer do not fulfill bounty starter's personal requirement. So what makes the post-bounty answer more deserving of the reward than its pre-bounty counterpart? If anything, the answer with the higher votes (regardless of whether it's pre- or post-bounty) make it more deserving than the other.
Now let's consider the scenario with reason 2. We don't really know the bounty starter's intention in this case. Maybe some post-bounty answer, or the (edited post-bounty) pre-bounty answer is what the bounty poster wants, maybe... but we don't know. And we shouldn't decide for the bounty poster to whom the bounty (even partially) is rewarded. This is just like we don't auto-decide that the answer with the highest vote gets to be the "correct answer" to a question – the question asker is free to choose any answer as "correct answer" regardless of how the community chooses.
To summarize, I'm arguing that if a bounty is not awarded within the stipulated time, there is no reason to award half of it to the highest-voted post-bounty answer. We can't and shouldn't decide on behalf of the bounty starter that that answer is the closest fit to what he needs, and in the case where it has less upvotes than a pre-bounty answer, it's not the best answer as selected by the community.
I say we should not award anything to anyone if the bounty is not awarded by the bounty starter within stipulated time. Same idea as we don't make a question asker choose a "correct answer" if he doesn't want to. I know Jabba wouldn't approve. The bounty starter would still lose the bounty though as they have had the benefit of increased visibility whether or not that led to an answer.
If we really must reward someone even though the bounty starter didn't make a choice, the answer with the most merit by itself (upvotes) should be awarded, regardless of whether it is posted before or after the bounty is put up, since we've already deemed that the bounty starter has forfeited his right to decide where his rep should go to. Instead of using the bounty starter's criteria, we use the community criteria – net number of upvotes.
[Bonus question: 10 marks] What is a bounty starter supposed to do if at the end of the stipulated time, he doesn't think any of the answers is to his liking? I think How does the bounty system work? is supposed to answer this question, but only thing that is remotely related to this is under the heading "What happens if I feel my question is still unanswered?" which explains that "automatic awarding" happens at the end of stipulated time, but doesn't say anything about what the bounty starter can do, things like marking it unresolved. Again, I've never started a bounty before so there might be things I'm missing.