WASHINGTON—President Donald Trump insisted that Republicans’ health-care plans would maintain a guarantee of coverage for people with pre-existing conditions—but sidestepped the thornier debate over whether proposed changes could raise insurance costs for sick people to the point where coverage is unaffordable.

“Pre-existing conditions are in the bill. And I mandate it. I said, ‘Has to be,’ ” Mr. Trump told CBS’s “Face the Nation” Sunday when pressed on whether the GOP health-care plan would maintain one of the most popular...