Do you ride upper level? Have you ridden changes well and correctly, with a quality canter before and after the change? Have you ever been described as a “talented amature”?
If you said “no” to any of these questions, you should let your trainer work on the changes, and you just work on developing the canter (because remember, 98% of the change is the canter before the change) until the changes are more reliable, because what you’ve read is correct. This can be a very emotional/challenging/difficult time for some horses and you definitely do not want them to be afraid, uncertain, or otherwise not confident about changes. You really, really want this to go as smoothly as possible for them because the changes only get harder as you go up the levels so you want the horse to be confident and the changes to be easy.
That being said, if you answered “yes” to all of these questions, consider the following. Whether its feasible for you to teach them (versus a pro who has done so on many other horses) really depends on 1) how easy your horse finds changes (i.e., do they do them naturally and correctly on their own out in the field?), and 2) how talented you are as a rider. If your horse has a natural aptitude for changes (i.e., does them cleanly on their own out in the field, or offers clean changes under saddle on their own), then you might be just fine if you stay out of his way. If he struggles, then he will really need you to help him every step of the way. Which brings me to my next point:
If you are a talented, upper level rider, you can probably do it. I did it with help from my coaches on the ground as you describe, but I had ridden lots of changes on lots of different horses, and have been described as basically a pro with a day job. I would only try them in a lesson until my coaches said to go ahead and play around with them on my own, and even then, it was only a few at the end of the ride a few times a week (never drilling!). If you are not a talented, upper level rider, you risk making the whole experience much more stressful and drawn out, and nobody wants that, least of all your horse. I was also ready at any moment to hand the reins over to a pro if I thought it wasn’t going smoothly (for my horse’s sake, this was not a time for ego!).
But remember, 98% of the changes is the preparation - months and months of developing the canter to be ready. So, you can definitely do that part with help from your coach.
Side note: if your coach doesn’t recognize how much prep goes into teaching changes, you need a different coach. Don’t let some “trainer” who hasn’t put correct changes on multiple horses experiment and learn on yours.