When user with participant, editors, privileged role clicks the scheduled reports they cannot see anything. But as a higher user they should see the cards through the scheduled reports.
Are the pages the cards are on or cards limited in permissions in anyway? You're correct, based on what you describe I'd assume they could click on them and go to the card. When you say "they cannot see anything" what does that mean? They get breought to a page with an empty screen? It doesn't go anywhere? Something else?