Aharanov makes two fairly robust assertions, does he not?
1) that our notion of time and space is fundamentally flawed
2) through preselection and postselection, we can read information from the future.
Those are his more outlandish claims, but there is no experimental evidence for those claims. His more central ideas are in line with QM and do NOT have information flow backwards in time. The Aharanov-Bohm effect has been verified, but it is NOT an example of information flow backwards in time.
And, in fact, one of the big problems in popular treatments of QM is exactly this sort of thing. The basis is usually trying to use classical mechanics and classical reasoning where QM applies.
For example, in the 'quantum eraser', it is common in popular treatments to see claims that there is information going backwards in time to 'erase' the results of an experiment. That is simply false.
Not sure why you have a problem with ‘popular’ treatments of QM. That sounds a little like the medieval church opposing translations of the Bible into popular European languages; lest the populace began to question the carefully curated religious orthodoxy of the time.
The main problem I have is that they get QM wrong much more often than not. They tend to emphasize the 'gee-whiz' aspect and the apparent paradoxes. They tend to say outright false things about information moving faster than light, or backwards in time.
I have absolutely no problem with people reading actual QM textbooks (which would be analogous to reading the Bible). What I have problems with is the poor quality of the popular treatments.
One of the biggest problems in QM is people using classical physics and classical notions when quantum mechanics is involved. They are different. And you *will* get paradoxes if you try to understand QM using classical ideas. But you don't understand the better description in terms of the worse one. That leads to bad results. Instead, you should understand why the poor description worked as well as it did using the better description.
In other words, the real question (which has mostly been answered) is why classical ideas worked so well for so long when it is actually QM that is what is going on.