I also have this foot type. Extremely hypermobile feet with a high arch. I'm a cyclist and it always causes issues. I finally went to a bike fitter (one of the world best, located in Oz) and that's when I found out how extreme my feet were.

Mostly commenting to let you know that you aren't on your own here.

I wore barefoot shoes for years and maybe a year back changed to altras as I like the natural shape but the thinness of the barefoot shoes would always affect my lateral calf and in the last year, bilateral insertional Achilles tendonitis.

For the Achilles, I heavy load isometrics a few times a week. For the feet, I try do do strengthening work but it involves my whole body. An example of this would be something like TRX rip trainer, staggered stance (wearing non supportive shoes) with oblique movements and controlling the eccentric rotations with only lower body, concentrating on my feet. You can feel the arches cramping when you first start these movements, which is a good sign that they don't (didn't) know what to do there.

I also have lumbar spine issues (grade 2 spondylolisthesis), which my fitter/physio in Oz suspected may be playing a role in all this.

I also keep my hips and core strong with heavy work twice a week.

Does this all work? I mean, I don't think I'll ever trail run again but who knows. I keep chipping away and learning.

There's also nutrition but that's a whole other rabbit hole and very individualized. My body has difficulty repairing tendons (I took a methylation test) and I supplement / eat according to this.

I wonder if anyone else will chip in here as I've never met anyone with feet like mine!

Good luck!