Russell did spend a lot of time on "the" in his Introduction to Mathematical Philosophy (chapters 16 and 17). But I haven't found a point in there where he makes a wry contrast between this pursuit and the ambitions of earlier philosophers to explain the meaning of life. On the contrary, he seems to take the matter completely seriously. From p. 167:

It may be thought excessive to devote two chapters to one word, but to the philosophical mathematician it is a word of very great importance. . . I would give the doctrine of this word if I were "dead from the waist down" and not merely in prison.

(He was in prison at the time for his pacifist activism during WWI.)