The article claims you can figure out the almost correct answer to the Monty Hall problem by running simulations hundreds of times instead of doing the maths (and the same for coin flips).

My problem is that it still relies on some mathematical intuition - that large sample sizes approximate the true distribution. Similarly bad intuition (like the gambler's fallacy) could easily be coded.

I agree that formally calculating the probabilities isn't necessary if you have the right intuition. But I believe getting good intuition is the result of training on problems (and then you can learn how to formalise it - which is the easier part).

Edit: Being good at mental arithmetic isn't necessary for programming, but being good at mental arithmetic isn't necessary for working as a mathematician either.

I don't think that there is any mathematical training needed to gain insight from running repeated simulations.

The only intuition you need is that you can become better at the game by practicing. This is a good (if optimistic) belief to have as a default. Then it's just a matter of playing over and over again and keeping score. It doesn't even have to occur to you that the strategy can be automated, you can play yourself. Just doing this you could build intuition for the best strategy the same way that most people can learn to play poker or dice.

You can say that the brain or learning process or whatever is obeying mathematical laws, or has learned a mathematical fact, but that's not the same thing as doing math or thinking mathematically.