Absent a physical or mental disability, the only people who can't be trained, are those who are unwilling to put in the work. That of course assumes the person providing the training is actually capable of providing good instruction.
I was once declared "un-trainable" by some military firearms instructors. Their instruction started, and finished, with "don't slap the trigger," and "cross dominant shooters are screwed." Funny enough, once I got some decent instruction, including an explanation of why cross dominance is irrelevant with pistols, I made rapid improvement.
Have you worked with the student in dry fire (preferably with a pistol mounted optic)? IMO, its best to spend some serious time working on the fundamentals in dry fire, before introducing the stress, and distractions, that come with live fire.
Are you sure its a "flinch?" I ask because a flinch is just as likely on the first shot, as it is on followup shots. Over the years I've seen things attributed to flinching, and/or the aforementioned, "slapping the trigger," when the problem was something else. What is the student doing with their hands, and eyes, as they fire the gun?the funny thing with this guy his first round is spot on, then it gets scary. I just think he can''t shake the flinch.
If you've diagnosed it as an actual flinch, and not some other problem, I think a true flinch justifies the cost of a Cool Fire system, to desensitize the student to the gun going off.
The noise, and "recoil" of the Cool Fire is enough to get the student used to the sound, and feel, of a gun firing. However, its much easier, less time consuming, and less expensive, for the student to use that 2 or 3 times a week, than it is to go to a range, and use live fire to get desensitized.