SAN FRANCISCO—Tristan Harris walked out of a sushi lunch and switched his phone out of airplane mode when it practically vibrated out of his hand. Ex-colleagues at Google and other friends were frantically trying to reach him.

Google Chief Executive Sundar Pichai had just announced that the search giant was going to give customers tools to help them limit the time they spend on smartphones. The idea was to help promote their “digital well-being.” In Silicon Valley, where keeping users online as long as possible has been the...