Solid recap. You covered Dock drag, the three dots, and app support limits. Gestures misfire occassionally. Safari, Notes, Files work. Some media apps block it.
Simpler route, use Stage Manager. Open Control Center, tap Stage Manager. Open app A. From Dock, press and hold app B, drag to the side until it snaps. Resize with the handle. This reduces timing wierdness. Example, Safari left, Notes right, steady.
If the toggle is missing, go to Settings, Home Screen and Multitasking, Stage Manager, enable. Older iPads lack it, check model support.