I would say that the reasoning behind a statement such as "good students are good in all things" is not one concerning the possibly wide variety of interests and abilities but one of emotional stability and general life competency.
No one in their right mind would doubt that different people are interested in different things and therefore not equally motivated to learn all skills and all knowledge. Most people would probably also agree that not everyone is equally talented at everything. Even the best students can be divided into those that excel at one topic and those that excel at another. There are rare individuals who excel in a lot of areas, but usually the best violinist is not also the best physicist, etc. So while "good" students are probably "good" in most areas, they are not the best everywhere. This makes the statement in question less severe, allowing us to rephrase it as:
"In most individuals, there appears to be a correlation between performance in different areas. Only very rarely is one person a top achiever in one field and a total looser in another."
The reason for this is that performance is, as we view "intelligence" today, not the result of an area-specific talent for, say, language versus maths, but that people differ in such characteristics as memory capaciy or processing speed, and that all skills depend on all of these physiological capacities. You need good memory for both language and math, you need processing speed for music as well as for physics. Therefore, someone who is good at maths is potentially also good at most everything else.
But more important than that all performance in all areas is based on the same physiological requirements is that the statement in question is, as I see it, not about ability at all, but about mental health.
What that woman is saying is that a person successful at life knows that good marks in school or college are important and has the will to get those good marks. Consistent good marks show not superior ability alone, but that a person has the emotional stability, familial support, and the physiological requirements to be successful in their lives in general.
If someone does not have straight As, with maybe a rare B, then you can almost always assume that they (a) are not good, or (b) have personal problems, or (c) are not interested. You want neither in an applicant. In that sense, a "good" (as in "good for our company") employee is (at least) "good" (that is, B) everywhere.