My first assumption would be just the opposite: that an approximation system (that or "ballpark" systems is what I call them, by the way) is just what somebody who has trouble approximating might need.
That might seem like a reasonable response. After all, it works very well with artillery.
But how many people do you see who miss shots by a mile--among people who play often and try to play decently (i.e., who might CONSIDER spending $500 for a lesson)? I think none. From my experience and observations, most people have no trouble getting close (say, within 4 inches or so of the pocket), but what they have trouble with is the higher level of accuracy. And that is also borne out in my calculations: A lot of shots require some impressive accuracy. From just watching a decent player miss shots, it could be difficult to determine what proportion is due to a problem with KNOWING precisely where to aim, and what proportion is due to a failure to HIT where he has aimed. The separation between competent players and "normal, just above bangers" players is in the fine details.
Maybe you mean that if one has difficulty approximating then he must also have difficulty fine-tuning? Could be, but that doesn't mean that an approximation system can't be helpful for both.
But then we have to ask whether CTE IS a good approximation system. I don't think it is (whereas GB CERTAINLY easily fulfills that requirement in a way anyone can easily manage--again, because most people approximate fairly well with NO help). If we use the 15,30,45 style CTE to start, then one must be accurate in naming the number of degrees of the shot before anything! Even the approximations in CTE won't be any better than the player himself can provide.
First: I don't see how CTE can even be much good for approximation--at least, any better than the user of CTE himself is at approximating. Indeed, when you get down to it, as far as I can tell, the player is actually fundamentally responsible for EVERYTHING in CTE.
Second: Since none of the CTE bells and whistles uses POCKET LOCATION in ANY WAY as part of its calculation (except for preliminary 15, 30, 45, perhaps--which the adamant poster AtLarge considers the HIGHEST blasphemy, and definitely NOT part of CTE), I don't really see how it provides any useful aiming direction AT ALL. Which leaves us with:
I think it's possible (in fact, I think it's probable) that some players' latent ability to aim by feel is "unlocked" through the use of these systems. It may be because the approximation gets them close enough to finish the job or because their belief in the system gives them the confidence necessary to do it or some combination or something else.
That might be a reasonable assumption, if the evidence were clear that lots of people felt strongly that they played much better after using it. But I've tried to emphasize that such data can be VERY TRICKY, and lead to wrong conclusions. Adding that to the fact that (I assert) it doesn't do anything ANYWAY...well, coming to reliable conclusions about what's happening with it is...problematic.
IMO, all that can be said with confidence is that you can find 15 or 20 guys posting on a forum who insist it has helped them...and, well, I gotta say...I don't find their testimony (or the way they present it) very impressive (some are, though). How do we know how many there are who feel it's done NOTHING for them, or just confused them? Such people VERY OFTEN are reluctant to stand up and be counted. Maybe they feel "stupid" because they couldn't "get it." Maybe they're embarrassed to admit it didn't help them. That sort of social phenomenon is well known. Surely you don't think ALL students are going to have magical, psychic transformations--and remember that you basically agree with me that CTE isn't going to actually tell people how to really make SHOTS. So, all those students who don't get a "subconscious overhaul" aren't going to be helped. Are you prepared to say that you know that proportion is minor?
What I know is that successful aiming relies to a great extent on largely subconscious processes that we understand only dimly - certainly not enough to assume that the many users who swear they get real benefit from these systems really don't.
Playing pool certainly is susceptible to subconscious influence. But I wouldn't call aiming "subconscious" as much as I would call it "conceptionless," or "wordless." We KNOW what we're doing, but it's not easy to describe (I'd describe it as aligning visual information with proprioceptive--until what our eyes see and our arms feel meet where wordless memory tells us has worked before).
Again, I'm just not as sure of the reports as others seem to be. Such things can be...really...a MINEFIELD.
I worry that such "interpretations" get started by people, like Dr. Dave for example, who doesn't want to make big stinks--so he works out a "fudge phenomenon" through which he can offer some acknowledgement. "Yeah, it helps people psychologically." "It helps people more consistently set up." That's fine to say; but, I repeat, what's difficult to KNOW is whether a significant proportion of people really THINK they've been helped, and whether those people are RIGHT about that (or are just improving over time, with practice).**
I'd be willing to make a sizeable bet (which, happy days, probably couldn't be pulled off

) that if you took large groups of interested pupils, taught group #1 good form and stroke + GB; group #2 good form and stroke + "cut feel" to practice; group #3 good form and stroke + CTE....
That the first two groups would do better than the third. That would be my honest opinion--and it's implicitly shared by instructors (such as Roger Long, Bob Jewett, Dr. Dave, Mike Page) who DON'T see a need to teach some version of CTE (which they could easily come up with if they wished, no doubt).
**And there's the "money phenomenon" too. Pay $500 for CTE, and I'll guarantee you that ALMOST ALL people are going to say it helped them (to say otherwise makes them look like idiots--to FEEL otherwise makes them FEEL like idiots). But if they bought it for $1.50 at Walgreens, maybe 95% would throw it in the garbage the next day.