An answer submitted before the bounty was started already cannot be automatically awarded the bounty This is confirmed by Jeff's answer here, with exactly the same rationale that you use:
It is not possible for an answer prior to the bounty start to be auto-accepted.
The assumption is that the bounty owner was not satisfied with any of the existing answers at the time the bounty was started.
You would have to award that answer the bounty manually, and I really can't see a compelling reason to prevent users from deciding to apply the bounty to any answer that they want.
This is especially true when you consider that bounties can be started on questions by users other than the person who originally asked the question. What they find most useful may be very different from what the original asker found helpful (and/or decided to accept). thanks Arjan for reminding me of this!
Also consider that some people might start bounties on their questions as a way of rewarding a second person who posted a helpful answer. Since the system only allows you to accept a single answer, people who don't think a mere upvote is sufficient use bounties to provide extra "bonus points" to a second answerer who was especially helpful in solving their problem.