The screens could definitely break, they were just very small so the likelihood that they would suffer an impact that would break them was comparatively small. In fact, the reason for the flip form factor was to protect the increasingly fancy displays when it’s in your pocket. They also didn’t weigh very much so they didn’t fall as hard.

Modern phones are extremely sturdy, people are just more precious about them because they’re much fancier and more expensive and more of a requirement for everyday life.