This feature is not actually related to secure DNS. You should feel free to keep DNS over HTTPS enabled as it will not interfere with this.
The best way to test it is to use http://https-everywhere.badssl.com/ (note that it’s an HTTP URL). When you visit that with Brave, it should be all green and the URL bar should show
https://. You can ignore the “Not secure” bit left of that, it’s just a consequence of how that test page is built. If you try the same test page in a browser without HTTPS upgrades (e.g. the stock version of Chrome), it will not be green.
Note that it’s not useful to install the HTTPS Everywhere extension in Brave since the same functionality is already built-in. Also, you can’t easily compare the two dashboards because they show different things.
Brave will show you when its built-in HTTPS upgrade engine has upgraded a resource from HTTP to HTTPS. There are however lots of ways for that upgrade to take place prior to that, for example via the standard HTTP Strict Transport Security functionality that many sites use.