It's not correct to say that AI researchers model the brain with Turing Machines (TMs), rather it's more accurate to say that AI researchers understand that the grammars of human language can be approximated with TMs. There is a logical equivalence between grammars and state machines and the thinking largely follows along the line that whatever it is about intelligence that allows people to use language allows us to trace back language and its mechanisms to intelligence. A militant belief in this is much less common these days with alternative connectionist models (models that mimic neuronal complexity and computation) being much more philosophically interesting; in fact, some researchers are trying to extend connectionism into the affective domain of human thought in ways that symbolic models cannot. Thus, while TMs are not models of brains, they share important properties of brains and can be used to create functional equivalences of the brain, so it is useful to compare and contrast them or use them to build connectionist and symbolic systems.
The Properties Brains and Embodied TMs Share
Philosophically, what AI/AGI researchers have been and currently are engaged in is an attempt to explain and characterize increasingly generalized forms of intelligence. I'm not aware of any assertions that TMs model the brain, but they can:
- Handle grammars like people can (uniquely in the animal kingdom)
- Are general purpose computers that can execute generalized action because they embody the same information processing cycle as people do; probably a derivative property of how grammars model the world
Thus, an embodied TM (the TM is an abstraction) represents two very important capacities, language and flexibility of action, that other animal do not possess. Thus, it is not a coincidence that computers like ENIAC which embody the von Neumann architecture have gone from doing simple arithmetic to classifying images of plants and driving cars sometimes better than people. The capacity for grammars, and therefore true language (as opposed to a system of signals) is fundamentally characterized by expressivity and that expressivity is fundamentally linked to generalized processing of information and action predicated upon that information. Now, in the early days, this was a good start, but fundamental challenges haven't been overcome. The dominant, or at least the vocal consensus, has been from the beginning that somehow language is not only essential, but is a pathway to the notion of generalized intelligence, a notion that has a number of competing theoretical models such as MI on the pluralistic front and the g-factor in the traditional psychometrics communities. But the question of what intelligence is is itself still very controversial among philosophers. The scientific approach is to establish a definition of intelligence through epistemic methodologies. That is the essence of an IQ test which indirectly purports that the measurement conducted by the test is de facto proof of the existence of the thing called 'intelligence'. Thus, a TM can do things that not even Koko the Gorilla can do. (Of course, Koko's lack of grammatical capacity is one of many intelligences that have yet to be built on a TM.)
Language, Intelligence, and Computation
In fact, Alan Turing's leap in his paper of universal renown was to establish an operational definition that measures language skills to make a pronouncement about intelligence, and if you read the paper carefully, what Turing claims is that given the electronic computers which might even need to learn in the same way as children, there's a functional equivalency between all agents that can handle a grammar equally well. When a person, under a game-like condition, can't really tell who the person is and who the computer is (yes, who), then computer is evincing human-level intelligence. To be explicit, no system to this day has done that, so there's a very solid theory in place regarding grammars, agency, and intelligence.
The earliest AI community was very optimistic that if symbols could just be entered and produced with enough quality and quantity, human-level intelligence would emerge in the technical sense. And some of the most famous pioneers in electronic computation put forth the physical symbol system hypothesis
A physical symbol system has the necessary and sufficient means for general intelligent action.
— Allen Newell and Herbert A. Simon
From this perspective, then, the question is how to do we make TMs manipulate symbols in a fashion to reproduce what humans can do. There have always been those who question this perspective, and certainly Hubert Dreyfus is one of the most famous of those upsetting many AI researchers with his view starting with a paper published at RAND and then moving on to expand those ideas with What Computers Can't Do. He raised obviously many of the same objections you have with enough academic reputation to deflate optimism, cancel funding, and discourage thinkers. There are still thinkers who believe in GOFAI, but these days, younger thinkers are encouraged that a long-standing thread in AI, that of connectionism, will add to the philosophical body of knowledge of AI to bring AI closer inline with the dream of developing AGI stemming from an artificial machine that somehow produces information LIKE the brain. And among those who know robotics, the claims have drawn light to the question of the inter-relation between the software and the hardware architecture which has deep philosophical implications. Today, such views are philosophically known as embodied cognition which says that symbols systems are fine, but they need to be tethered to some body in the physical world so that transduction can play a role in machine learning. The result has been that over the last 20 years, AI researchers have focused on methods for supervised and unsupervised learning which, instead of attempt to instruct a computer like a child is instructed using rule sets and language (think expert systems for instance), exposure of connectionist models to raw input condition the machine to develop a sense of normativity. Thus, a computer can now read your cursive, decide if your pet is a dog, decide the fastest way to ship a package to you, and drive you to work, none of which require explicitly encoding domain-centric rules (though a synergy is superior) resp., handwritten allographs, image recognition, logistics, and transportation; these connectionist and hybrid models instead rely on being exposed to similar contexts and then abstract what can only be describe as intuitions. Honestly, it's not unfair to characterize machine learning systems as possessing intentionality in a primitive sense, though many thinkers bridle at the suggestion. But whether one leans towards symbols or connectionis as a model for imbuing intelligence, both can be implemented on an embodied TM. Thus, brains and TMs can be shown to have some very powerful properties in common.
Today, TMs are tremendously important for understanding how to model the functions of the brain, but for reasons as you have discussed, many researchers believe that the TM is only part of the necessary design of a system to manifest greater and greater levels of intelligence and are not sufficient. TMs are algorithmic and human minds often are not. Most AI people I've known or read think that there's a symbol-connectionist synergy at work, and that new generations of AI technology will continue to employ both strategies. The brain, after all, is not a von Neumann or Harvard machine, but is this object conjoined in a fantastically complicated way with a body, and that both are connectionist in ways that simply that we can't hope to deliberately articulate with grammar. You could use a TM to encode rules to recognize faces, for instance, but that's very difficult. Instead, facial recognition uses the TM indirectly and builds up an abstraction to allow for machine learning from physical inputs in a way that our intuitions about people's faces might inform our conversations about them.