I think the media has a point in that a presidential candidate needs to be more right of center to appeal to a broad range of voters. Extreme ideologues on both sides have gone down to crushing defeat in the past (Goldwater in 1964, McGovern in 1972). And, it depends on what is meant by a strong conservative.  A Ted Cruz would have a difficult time appealing to a broad swath of voters while a Marco Rubio or a Chris Christie might have broader appeal. It's not the strong conservative bit that is problematic; it's being a smart conservative.