I don't think it will become a serious concern. At current, each user can only post a bounty on one question at a time. Certainly, there may be a lot more bounties posted in the first few weeks of the new system. But I have a hunch that over time, it will dwindle down to the not much larger than how it currently gets. Larger, yes, but not dangerously so.
There may be more bounty questions around as a result of this change, but all of them are still separated from the gigantic question pool of all questions. I don't expect it to become so large as to be unmanageable. It still costs reputation, and it costs more to be as attractive as before now that the bonus +50 is gone. So the people who were worried about the final-ness of posting a bounty and the risk associated (like me) will be more likely to post a bounty, but the many people who are concerned about their reputation changes will still be very hesitant to post.
Now, the most likely scenario with the new system is that you will see a lot more cheap bounties. So post your bounty with more at stake, give greater rewards! This new system actually makes something like this feature request more valuable. Beat quantity back with quality! Show people that your question is worth their time!