I'm not a big Yoga person so my answer will be different. Why would you want to shut your brain off?
What you want to do is be able to slow things down, like Neo in The Matrix:
You want to be able to slow down time little by little so you can see the water falling, and can catch a fly with chopsticks. But to turn it off - why?
Over analyzing is good. The problem with Woody Allen is that he didn't really go over those basic stages (though his latest film is great), but when one is good at analyzing one can also move on to greater mountains.
You say Robin: "The danger of the over-analyzing person is that he forgets his place in
the here and the now."
Not at all. If you over-analyze in a good way, and in a constant way, then you can get to be very good at it and see things very quickly and analyze them on the spot (as you say you can do). The here and now become infinitely richer like that as so much more is going on than first meets the eye. The Matrix is again a good example.
I don't know Yoga at all, nor meditating, but I think a lot of what is meditating in a deep way, like the martial artists monks did, is mostly misunderstood today. You want to be able to feel the water going through your hand, feel the air touching you, hear it move, but not at all to be numb. Not at all to turn the mind off.
(Sportsmen say that at times when they are completely in-the-zone, that everything moves slowly for them.)
"...a lot of what is meditating in a deep way, like the martial artists monks did, is mostly misunderstood today"
There's a remark somewhere that the warriors of old were able to clear and focus their minds so completely in the present moment that their senses became incredibly magnified: walking along on an ordinary day, they could hear a leaf drop hundreds of meters away, and the slight footfalls and rush of air that meant someone was about to turn the corner several houses back. Only with this kind of focus could one survive the danger of attack at any moment.