For what its worth (and having done a bit of study on this and being really interested in the topic): the answer seems to go back to the beginnings of AI and even earlier (Turing's 1936 paper in which he introduces what's now called the Turing machine).
John McCarthy's filer for the 1956 Dartmouth College summer workshop on "Artificial Intelligence" (which name introduced the term "Artificial Intelligence") in part says:
"The study [workshop] is to proceed on the basis of the conjecture
that every aspect of learning or any other feature of intelligence can
in principle be so precisely described that a machine can be made to
This references Turing's 1936 paper where a machine or natural system is described, and the description is run in a computer. To simulate is to quite precisely describe a system then run the description (transformed a bit – but the result is still a description) in a computer. The description is the program. The description needs to be precise, as indicated in the Church-Turing thesis.
So the idea of simulation is core to the computational theory of what a digital computer can or might do. So it's also core to the computational theory of mind (the organic brain being a natural system), and hence to AI.
That said, it's obviously a crazy idea to try to quite precisely describe the organic machine that is a human brain. I mean how many neurons? 100 billion. Quite precisely describe each and every single one of these, and each and every of the up to 10,000 connections that connect to each and every single neuron. Crazy with a capital C. And to suppose there are degrees of simulation of the brain, or that the mind is somehow a simplification of the brain, or that the description can be in higher level concepts, not neurological ones, is just to admit that the description is not quite precise. An adequate simulation of a brain would be terribly detailed.
So why do we hear so much about AI trying to simulate the brain? Answer: AI has no other word to express what it does.
In my view, AI ought to be trying to work out the data-processing principles of the organic brain, not trying to describe the causation of the brain. AI doesn't know the principles of perception or the principles of general knowledge. It's incredible to say this – seeing as both are so absolutely fundamental to human intelligence. But AI doesn't know the principles. It ought to be trying to work them out. Then – once discovered – to work out how these principles could be realised in a computer.
You suggest that there's a binary choice between AI trying to get a computer to simulate the organic brain, and trying to grow organic brains in a dish. But there's actually a third option. Computer can do things other than simulate (i.e., other than compute). Maybe these other things might include embodying the principles of organic brains.
There are two really big areas here: (1) what are the principles of intelligence? (2) what are the non-computational things computers can do?
You ask why AI is concerned with the digital environment rather than, say, growing organic brains in a vat. But AI is basically an engineering project (building something with a designed causality) and even though AI knows only a little about the causality of what it's trying to build, the digital computer seems to be the only viable platform, at present, with enough individually addressable memory locations and processor speed to cope with semantic structures that would result from an adequate sensory interaction with the environment.