Excellent advice!
AssistiveTouch sort of worked (or I'm using it wrong), but the "double-click" and "triple click" feature left me waiting for 1 second or so until I was returned to the desktop again. Very strange. What worked was pressing the virtual home button. A downside to using this feature is that when taking screenshots the "black square" will be shown in the images.
Using a 4-finger gesture however seems to have solved my challenge! If I swipe with 4-fingers (5-fingers is too tricky) from the bottom of the screen and upwards I get to choose between the apps in use (same as triple-clicking the home button), and from then on I can select the app I want, quit an app or select the desktop. Very useful
I do hope the physical home button will last for a while, but having experienced an iPod Touch with a worn out home button that needed replacing, and ending up wrecking the entire device when trying to replace it (very fragile wires, connectors etc. apparently not meant to be fixed but thrown away) I don't want to repeat that with this iPad Mini.
I quit apps because of limited space (16 MB), so as to make it snappier. No need to run 10 apps at once