Yes, a closed question cannot have any bounties; conversely, it's impossible to close a question with an active bounty. AFAIK this is because bounties are mainly aimed at getting new answers; almost all text in the help center articles seem to imply this.
When should I place a bounty?
In order to get good answers, you have to put effort into the question. Edit the question to provide status and progress updates. Document your own continued efforts to answer the question. This will naturally bump the question and get more people interested in it.
However, if after doing this, you feel a particular question still isn't getting enough attention, adding a bounty may help.
(from the privilege page)
That you can award an existing answer is just a nice bonus, but not essential to the bounty functionality.