OK, I'm not an expert. My feelings on these matters is only based on my own bad experiences, and talking to others who have had problems.
From what I have seen, the best skate fitters do look at things like flat feet, pronation, etc., and mark measurements relating to them on the fitting forms. It's pretty common to also do a foot impression on foam, that helps tell the manufacturer what footbed shapes and orientations are needed to make the boot work right. I don't know how common that is on semi-custom fits.
But even if you order full custom boots, having to do minor fixes, is pretty normal.
Doing those fixes is part of the job, as I view it, of a skate tech. He/she should do whatever is needed so that wearing and using the skates creates no pain, no sores, etc. IMO, he/she should not cause an injury to his/her customers, as you think yours did. (That said, strength, endurance and flexibility training all create various types of dull pain, sometimes delayed pain - but that pain different in character from the type of pain that indicates that harm is being done.) A day or two should be all that he/she should need at most, if he/she knows his/her business. If he/she can't do that, I think he/she should refer you to a medical professional who can, and who has a lot of experience working with skaters. But obviously you disagree, and are more forgiving of your tech.
In fact, not all skaters in my area with such problems learned to do things themselves, though it is astonishing how many do, and how many coaches get fairly good at it too. But a lot of the skaters in my area ended up going to a particular podiatrist who developed a good reputation for dealing with skaters.
I think you shouldn't need unbalanced muscle use to stand on one foot. If the skates are done right, you should be able to glide on one foot with normal symmetric muscle use.
If you do feel that you need to strengthen your core strength to balance on one foot better, you might try a variety of exercises. A PT could show you a bunch of such exercises, and could also help you figure out what issues you might have to work on. Your coach, if you have a private coach, may be able to do this too.
The most obvious exercise is to stand on one foot for a while, off-ice. Tighten your core muscles to create balance - e.g., in the abdomen and lower back, and if needed, your lower oblique muscles. Then do it slowly moving the free foot around to various positions. (E.g., the performance dance community likes to move the leg between front/side/back/side positions, at various heights.) That strengthens different groups of muscles, as the leg moves around.
You can graduate to a balance board, like this cheap one:
https://www.fivebelow.com/series-8-fitness-trade-workout-balance-board.htmlor a more expensive "half bosu" ball, with or without resistance bands which might help you balance:
https://www.amazon.com/URBNFit-Balance-Trainer-Stability-Resistance/dp/B06XZYX5QD/ref=sr_1_1_sspa?crid=2IC0OKGYJ4402&keywords=bosu+ball+balance+trainer&qid=1568039258&s=gateway&sprefix=bosu%2Caps%2C133&sr=8-1-spons&psc=1&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUExRlBPV0JFM0dTN1g3JmVuY3J5cHRlZElkPUEwNDYzNzI1MTFXQ0RYUklZVDc5TCZlbmNyeXB0ZWRBZElkPUEwNDEyMTk2MVJKSjZGTjBCQUdNWSZ3aWRnZXROYW1lPXNwX2F0ZiZhY3Rpb249Y2xpY2tSZWRpcmVjdCZkb05vdExvZ0NsaWNrPXRydWU=Many gyms, BTW, have something that looks like the second one, though usually without the resistance bands, that customers can use.
I use it with the rounded side down, put one foot on top, and try to stay balanced. I also make sure I am close to something I can hold onto, so I don't fall off. Then I try slowly moving my free foot around. I'm sure your coach could suggest better training exercises. Some people do it on two feet, and do deep knee bends, trying to stabilize their orientation as much as they can.
The cheap one I mentioned doesn't let you go off balance as far, so you are less likely to fall off. I prefer it. Though it might not be strong enough for a heavy person.
There are a whole bunch of Yoga exercises that focus on balance and core strength too. E.g., "Tree pose". The exercises are all easier to learn off-ice, rather than while gliding on-ice. They are nearly impossible balanced on a skate while standing still, so don't worry if you can't do that.
But you can always come to your own conclusions and solutions.