> There are no standard integer types that take 15 (decimal) digits to represent

Nitpick: that is irrelevant. The code reads in at most 14 characters.

> Why would you use fgets() instead of just gets()?

You don’t use gets because it doesn’t exist anymore. It got removed in C11 (it rightfully was deemed so bad that backwards compatibility was sacrificed). You can use

  char *gets_s( char *str, rsize_t n )

, though.

(https://en.cppreference.com/w/c/io/gets)