Previously I could easily select the object behind by just clicking twice (or more if there are more layers), but now it doesn't seem to work. Moreover, I can't select smaller objects at all (only via layers panel on the right)
No, 3.0 didn't change that aspect of the selection process. However, there is an option in preferences OpenGL Depth Picking that influences it. To check whether you have accidentally disabled it, Edit→Preferences; select Viewport and see if it is checked:
If the option is not set, then selecting an object behind another one fails.