So after hours and hours of struggling, installing version 2021 and 2023, uninstalling everything, reinstalling 2022.3 LTS from scratch, I accidentally realized that my touchpad (Dell XPS) has two ways to right-click: tapping it with two fingers (which I’ve been doing for years) and pressing the right-bottom corner (which I ignored since the beginning). The first way doesn’t work in the Scene window, while the second one does, and I am able to rotate the view and move in flythrough mode with WASD.
Now, I checked other windows of the editor (Hierarchy, Project), and they accept both ways to right-click with the touchpad. So there is definitely something fishy about the Scene window, and I do think it’s a bug in the editor, but at least I am not blocked anymore.