Great discussion!
My own take on this is that one is greatly helped by having his or her “SAT CAP” on when approaching the test. This includes the need to abandon what I would call the standard approaches described above and the usual … reflexes.
The key, IMHO, is to take a couple of seconds and look at the question and the suggested answers before marking “something down.” I think that the first entries on a paper are extremely important. If I look at mine in this example, my first entry served to eliminate the division.
(4-x) / (2+x) = x became 4-x = x^2+2x
Obviously that led to the next 0 = x^2 + 3x - 4 as I always like to test a zero answer. 
As always, there are no hard rules. The type of “gymnastics” does, however, come from practicing, and often from repeating the problems one DID solve but only after wasting precious time.