One of the key aspects of the "Liberal Arts" is that they liberate us to learn and develop on our own without further input from a teacher, by learning to judge our own thoughts (effectively becoming our own teachers). This allows us to progress beyond the currently known into the unknown; it allows us to challenge the existing assumptions of a given field of expertise; and it allows us to join multiple fields in new ways.
I would also state that this is the difference between "Training" and "Education".
Having a broad understanding of a lot of things might make for some nice, simple dinner discussion, but if we ever strive to understand anything at all, we need deep knowledge. That's why we have different fields of study and career. I personally am not satisfied if I have a general knowledge of something that interests me. I love researching into some things, and others I couldn't care less about.
If you could learn how to do pretty much everything, then I'd say yes. But since you will always be dependent on others for something, it makes more sense to provide one or two services and do those really, really well. Note how it says "things" not "thing" so it's plural- a couple of things that you can provide really well is better than being able to do several things at a mediocre level.
School encourages students to do well in every subject. We are not rewarded for going beyond the given lessons. We are only rewarded for studying what everyone else does, and scoring high in those tests. We should encourage students to 'go beyond'.
Having a deep understanding of a specific field let's us venture into the realm of the unknown. What I mean is that the more we focus on a specific field, the more efficiently we advance. For example, we could get all A's in school, but if you do not find passion in on specific subject/field, you won't advance efficiently. Instead, you will pull in every direction, thus not advance. Instead, if we focus and contribute all our power to one specific field, we will advance out of the realm of the trivial 'common knowledge'.