I am against removing the automatic doubling of bounties. When original bounties are not awarded, I feel it would be especially counterproductive, for two reasons:
If an initial bounty did not attract any good answers, a new bounty should be for a larger amount, to work better towards attracting answers.
If you posted an answer, and did not get any good answers, we already have evidence that the question is either too complex, or otherwise too difficult to attract good answers. I never see a bounty question completely ignored; if I see a bounty that includes no answers, I still see a select group of users trying to work out a good, canonical answer, in the question comments. So people still put in the effort, but sometimes, a question is just simply too difficult to answer. Sometimes, the bounty still does not justify the required amount of effort or research to establish a good answer.
In such cases, doubling the next minimum bounty amount is not only more efficient, but it is more justified; If users still have trouble answering the question at 50 bounty, a good answer for the question is more likely to deserve the second bounty of 100. At the very least, you have to offer something else to generate traffic. Einstein's definition of insanity is "doing the same thing over and over again and expecting different results". At this point, posting the same bounty when the said bounty has already been posted unsuccessfully feels counterproductive. We have already established that the question is difficult to answer; we should be trying more to incite a good answer.
Doubling the required bounty is a feature partly intended to limit "bounty abuse".
As pointed out, in the comments, the feature of minimum bounty doubling actually aims to reduce system abuse by continuously posting bounties on a question. Long story short, a user could theoretically continuously bounty their question to ensure it is always under the spotlight, and unfairly game reputation.
It seems like the cases where the original reward was not awarded seems especially likely to fall under this case; especially if you include "not awarding the bounty" as cases where the original poster does not manually award a bounty, but an answer receives enough votes to automatically receive half.