sizeof is a C keyword. It returns the size in a type named size_t. However, size_t is not a keyword, but is
sizeof is a keyword because, despite it's name and usage, it is an operator like + or = or < rather than a function like printf() or atoi() or fgets(). A lot of people forget (or just don't know) that sizeof is actually an operator, and is always resolved at compile-time rather than at runtime.
The C language doesn't need size_t to be a usable, consistent language. That's just part of the standard library. The C language needs all operators. If, instead of +, C used the keyword plus to add numbers, you would make it an operator.
Besides, I do semi-implicit recasting of size_ts to unsigned ints (and regular ints, but Kernighan and Ritchie will someday smite me for this) all the time. You can assign the return type of a sizeof to an int if you like, but in my work I'm usually just passing it straight on to a malloc() or something.