@acidinmyfridge Thank you for your suggestion. I actually tried Vivaldi already, and neither with it does Duolingo's speaking exercises work. Duolingo web page's settings, those on the page itself, don't even show possibility to turn microphone on or off when using Vivaldi, even though I've explicitly allowed it for Duolingo in Vivaldi settings. This is true even when running in incognito mode with no extensions at all.
In fact, with Vivaldi Duolingo does not even give speaking exercises, since the browser apparently can't even see or understand my audio input devices, and thus the website thinks I have none. So no luck there.
I've already switched back to using mainly Chrome, since everything works smoothly there. It's a horrible memory hog though.