I really think that a very simple gradient or a 1 pixel wide white/black(depending on background) border around the edges of all interactable elements would go a long long way to fixing this issue without breaking up the UI too much, it might even expand it because just having that distinction gives you another "word" in your visual vocabulary.
Buttons do have a border. The tiles on the start screen have not. Mostly because they're all tappable (but probably also because the background provides separation and thus a border on its own). There are no non-interactive parts on that screen.
*edit: Changed "buttons" to "elements"