Where do transformational grammars stand in the Chomsky Hierarchy? From what I understand, natural language are too complicated to be generated by a context-free grammar, yet parsing a context-sensitive grammar is too computationally demanding for our brain to be continuously performing. So it seems that if our brain indeed has a "syntax" module to parse sentences as Chomsky claims it does, then whatever it's doing should sit somewhere between CFGs and CSGs in terms of computational complexity.

I'm really not familiar at all with Linguistics in general and even less so with Chomsky's work, but (correct me if I'm wrong) the point of transformational grammars was to explain this?

For those not familiar, a transformational grammar consists of a CFG to generate syntactically valid sentences, and also on top of this "transformational" rules, that transform these valid trees to form other valid trees. A common example is affirmative->question transformation ("John has eaten all the heirloom tomatoes." -> "Has John eaten all the heirloom tomatoes?").

It is not clear to me why such tree transformations couldn't be integrated into the CFGs by adding more rules (for example, question-form sentences can clearly be generated instead of being transformed from an affirmative sentence).

If I am right (in that transformational grammars sit between CFGs and CSGs), could someone give me an example showing both relations (that TGs are more expressive than CFGs but less expressive than CSGs). If I'm wrong, then please help me increase my knowledge of linguistics by pointing me towards valid readings.
I was taught in linguistics class that Chomsky abandoned his original arbitrary transformations after they were found to produce Turing completeness. See e.g. the Oxford Handbook of Linguistic Interfaces, p. 543; I am unable to locate the original reference.

He subsequently developed a formalism with more restricted transformations; the version I was taught was called Government and Binding Theory.

Some details are in Wikipedia under Transformational grammar.

While searching for references, I ran into a highly readable article on this matter by Geoffrey Pullum, which rejects Chomsky's original notion of transformations for purely linguistic reasons.