Maybe inherently was the wrong word. Though you can imagine a logger API that was like this:

    typedef enum {
        CONSOLE_LOGGER,
        DATABASE_LOGGER,
        /* etc. */
    } LoggerType;

    void add_logger(LoggerType);

In this API you're constrained by what loggers you can add. It's not the totally unchecked free-for-all that a callback-based API provides. The user is strictly unable to shoot themselves in the foot.

But this limits the expressive power that callbacks provide. Sorry I don't have any other API recommendations. The only way I can think of to stay expressive while safeguarding against unintended abuse is to include code analysis.