Something I thought of, not sure if it is any better than giving your beta readers an iTunes checklist, but when I was working with someone’s draft I gave them my Kindle device email address (I think all readers and devices can be set up with one). Whenever there was an update to the manuscript, she would send the .mobi file to that address, and Amazon sorted out the conversion and delivery to the device. So if you had them set up addresses like that, you could BCC updates to them all and it would theoretically all be pretty automatic.
I hit this issue today and googled and came to this post. Works on the Mac preview and not on iOS devices for mobi compile like everyone else (emailing it to kindle emails because I’m trying to use that to send to some early readers). Thanks to Cynthia for her post. I did it even simpler, I compiled to EPUB from Scrivener, and then covered that to mobi using Calibre. That rendered fine it looks like right now, but I need to comb through in more detail and see if it picked up all the formatting like block quotes and such.
Two and a half years later and this problem still exists. Thanks CynthiaBlue44 for the documenting the workaround, it’s still necessary as of September 2021.