Why do North American law schools require an undergraduate degree, when the UK's don't?
Note that a US undergraduate degree is very different from a UK degree. The latter is much more specialized. In UK it is assumed that the general education occurs before the university level. 

But in the US, the degree is quite general. Students of, say pre-law also study things like history, philosophy, foreign language, etc. It isn't that different from a History major. The "major subject" in the US accounts for half (more or less) of the total credits required. Even a math major will study all those things in the US. 

And, in both places, it is generally believed that a lawyer needs that broad grounding. It is really just a question of where and how it is obtained.