You make lots of comparisons to golf, a game which I never played seriously. However, I do see this inability to really see what is going on with my fly casting students.
Most folks see a guy making a forward cast with a fly line and to them it looks like you are throwing the line out there, like you do with a weighted lure or sinker. The action looks similar to spin casting, but the physics and biomechanics are completely opposite.
For one, the fly you are casting weighs next to nothing compared to a fishing lure. With spin casting you are using the leverage of the rod to accelerate a weighted object to high speed. When you release your finger from the line at the correct time, the heavy lure goes sailing out to the target, dragging along the almost weightless monofilament line with it.
In a fly cast, you throwing a weighted line to drag along an almost weightless fly. The line is propelled forward while the fly just goes along for the ride. The main difference is in how the line is accelerated, and that difference is in the timing. It is a
felt thing (as in requiring the sensory input of the fingers, hand, arm, shoulder, hips, legs, and feet) and is very difficult to break down and describe like you did with the golf swing. If you blow the timing, maybe push a little too hard or not hard enough through the stroke, the tip will not travel in a straight line and you will get very poor results.
The hardest habit I have to break is with guys who have done a lot of conventional bait or spin casting and have the habit of applying a lot of force at the end of the casting stroke. With a fly cast, the line has already accelerated as fast as it will go by the time you are ready to finish the stroke. All you are supposed to do is to stop motion as abruptly as possible. This causes a tight loop to form in the line, and the rest of the line (already in motion, but behind you) just rolls over the top until it is straight, then falls gently to the water.
Muscling the cast at the very end not only fails to improve the cast, it actually kills it. And yet, I will see over and over a guy finally getting down the timing as he does his practices casts (termed "false casting") back and forth before the final forward cast, only to ruin the cast by attempting to throw the line like a regular lure at the very end. By doing this, he tightens up his entire upper body and arms and then lets loose at the end. What he should be doing is the exact opposite. Deliver a forward cast exactly the same as all the practice casts - with a fluid and relaxed forward motion - and then to slightly firm up the grip at the end of the stroke, effectively inducing what some call a "power snap" (yes, a very unfortunate choice of words, I never use that term with my students), which stops the rod and allows the formation of the essential loop.
Sound familiar to you, CJ? It should, because it very closely resembles the final wrist action in your pool stroke.
Anyway, this is just one example I could think of where things are very different from what they appear to be on the surface. And just like in pool, some guys can be very resistant to attempts to correct the stroke. I tell newbies to just feel for the "tug" - the pull of the line as it reaches the end of it's travel - before they attempt to change direction of the line. Best case scenario if they don't is they end up with a "tailing loop" which kills the cast completely and causes it to fall to the water beside you. Worst case is like with my buddy, who after watching me for a while, got it in his head that he just needed to "wave it back and forth" and ended up with 40' of fly line tangled around his head and a size 12 dry fly embedded in his ear.
Lucky for him I always use barbless flies.