Update:
Please go to brave://accessibility/ via your address bar, and enable the following:
- Native Accessibility API support
- Web accessibility
- HTML
Upon enabling these, please restart Brave and let us know if this improves your experience.
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First off, what a top-notch report, @Mike.Antonacci 
I assume you’ve assisted projects in the past with understanding some of these a11y issues; I’m blown away at how well you’ve put together this feedback. Thank you!
Second, Dragon Naturally Speaking is really impressive. I first used it back around 2003, but not much since then. What a11y I have done over the years was largely around screen-readers, and making sure they were able to track context and more on pages. Out of curiosity, how much time did you have to spend training the software on your voice? If I recall correctly, back when I first used it, it was only a few minutes of reading a short story—curious if that has changed any.
Regarding the issues you’ve captured here, is this something we would be able to reproduce without a paid account? If not, would you be willing and able to join one of us over Zoom for a remote-troubleshooting call? I’m curious if Dragon leverages an in-browser extension, or if it operates exclusively from a parallel program running on the same machine.
Because the software works better in Chrome than in Brave, could you provide screenshots of the about://version page from both browsers? I’m primarily interested in the “Command Line” portion, which looks like this:
Lastly, I was curious about Dragon’s own settings. I noticed when you were dictating input into the pop-up box, that Dragon was configured to Use automatic Dictation Box for this application. There was also a 
icon at the top-right; can you explain these settings, and share what other settings exist for per-application interoperability, please?
Again, if these are questions we can answer on our own by using a demo of the app, please let me know. I’m more than happy to download the software and work towards making routine testing a part of our regular process.
Thank you again for the excellent report; I hope to see this matter resolved for you soon!