Reading and language are not the same thing, e.g. you can speak a language without being literate. You might therefore expect different writing systems to use different capabilities. Of course, "different parts of your brain" does not mean different capabilities. They could be the same capabilities instantiated differently in the brain.

It may very well be that speakers of a language with feature A can perform some task better than speakers of a language without feature A. However, it does not meant that the speakers have different cognitive capabilities (e.g. you can provide sufficient task training), which is the ultimate argument of people supporting linguistic relativity.

I cannot comment on the political motivations to deny linguistic relativism. I am just a linguist.