Is it possible to enable bounties on questions tagged puzzles to be set immediately? eg; I post a puzzle and then set a bounty on it immediately after posting?
How do people feel about this as a "Feature"?
This is not possible, and I doubt the company would enable it. It's simply not what bounties are meant for:
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.
On our sister site Puzzling Stack Exchange, users usually post hints if a puzzle isn't solved within two days, and seldom resort to bounties.
Also, posting a bounty on new question doesn't mean much in terms of visibility; right now, the homepage covers almost a week of activity (when set to the maximum of 50 questions) or almost two days (when using the default).