It would be hazardous to allow bounties to be awarded back to the requester. Consider that this would basically fully refund the bounty, meaning no net change in reputation. Subsequently, the requester can request a new bounty on the same question, putting it back in the Featured questions for another week. The lack of award not only mimicks the original system, but also helps prevent gaming the bounty system.
The lack of a record is probably derived from the fact that there is no real change. I mean, the fact that a bounty was ever assigned to a question is never pointed out at all, but there is a [status-planned] feature request to circumvent thisthere is a [status-planned] feature request to circumvent this. So in line with that, since no bounty was awarded, there's nothing to mark on an answer. It's basically discarding the bounty.
If this lack of record is the bug, then highlight that as the issue instead of the ability to self-accept. Because I don't see the ability to self-accept being a problem. I agree that the lack of record is bad, but I don't see any harm in letting us self-accept. Nor do I see that converting the current process to some "discard bounty" option will technically fix this. All that does is change the name, you still need to have some method of recording that. And if that isn't recorded either, then there's no gain at all from getting rid of the ability to self-award.