If you want to see dead ends other people have gone down and read some actual results and learn some history it is a good read for the mathematically inclined. Some of the maths went over my head but I enjoyed it!
If you look at this as a series of consecutive operations, every odd operation grows by a little over three, and is guaranteed to be followed by an even operation. Even operations shrink by a little more than three, so all that's required is to compare the cardinality of those two offsets to see whether continued operations are growing or shrinking. In the case of odd operations, as the infinite series of operations continues, the offset gets smaller and smaller. In contrast, the size of the offset for even operations doesn't change as the series of operations continues, which should mean that numbers shrink more than the grow, just ever so slightly.
(To understand the even operations, you have to look at expected values. 1/2 of all even operations will produce an odd operation after dividing by two, 1/4 after dividing by four, 1/8 after dividing by eight. The infinite sum (1/2)^2n converges to 1/3, with an actual value of 1/3 - (1/3)*(1/2)^2n.
1. It doesn't show the result for all natural numbers, only some "high probability" fraction of them (similar to what Tao did).
2. The expected value of the ratio being 1 doesn't imply that the actual number goes to 1, since it could be stuck in an endless loop with itself as the minimum.
3. 3n+1 doesn't necessarily have a uniform distribution over the even numbers. Tao gets around this by picking some "stable" subset of numbers that don't move too much under the transformation.
It is an extremely persuasive one, however, and and as result of it, nearly all mathematicians believe the conjecture is true.
The persuasiveness of the heuristic is one of the things that makes the problem so confounding. It's obviously true; it's silly to argue that it isn't.