I may not answer for Mathematics specifically, but I would be generally less pessimistic about the politics in academia. But there are many layers to peel here.
First, "do not try to challenge famous researchers" may be a useful heuristic. They became famous for a reason, and lots and lots of resources were spent following in their footsteps. Challenging some of their results without a strong attack is unwise, and going after foundational principles of any field will cause an instinctive rejection by the community, as it have many times in the past. On the other hand, no guts, no glory - if you DO have a strong attack on a problem, it may very well be the road to fame. Planning research around having a breakthrough is not particularly feasible, however.
Second, in some parts of the world, challenging the authority has broader cultural implications. This is far from universal, but (allegedly) there are places where it may well be a career suicide.
Third, if the previous point does not apply, I would call "giving very much credit" either a poorly worded or poorly understood (either by you or a person who stated that) approach. See, "famous authors were all wrong" implies that anything based on that paper is invalid, and that stance loses immediately. Instead, try to reconcile the development of the field with your newfound results. It may not be that important to pay tribute to specific people, but failure to recognize the value of a significant fraction of research done in the subfield would easily lead to the paper being rejected, ignored, or even labeled as crankery. Starting off by antagonizing the very community receiving this work is very counter-productive, and not being sufficiently specific with the critique may, unfortunately, trigger certain heuristics in fellow researchers that will place your paper straight into the trash bin. "Giving credit" may have the same presentation as "reconciling the differences", but a very different driving force behind it. The latter tries to be impartial even towards oneself, the former makes scientific merit highly subjective and, plainly speaking, feels wrong.
And finally, there is certainly a component of having to work around personalities in academia. The higher up the ladder, the more political it gets. In that sense, yes, famous people wield a lot of power, and if you try to get a big project going, you have to consider who will be in the committee, what they believe has merit and what does not. There are not enough funds going around to pursue every venue meaningfully, and attempts to divert more resources than just venture capital from what seems to be working great otherwise would not be received favorably. Convincing other researchers that what you are working on is important is a substantial part of being an academic, and this is especially significant in mathematics as this acute observation by Alexander Woo attests.