It serves a great deal of purpose, if you accept my logic as to the purpose of the pocket veto. Allow me to explain, by imagining the Constitution without this clause.
A presidential veto, the framers determined, must take the form of a communication to the house of Congress in which a bill originates. (They borrowed this from the New York and Massachusetts constitutions, which pioneered the two-thirds override.)
And the President has a ten-day time limit in which to act, after which the default is for the bill to become law. Otherwise, the President could sit on a bill that he didn’t like, forever.
Given those two provisions, and without the pocket veto, Congress could subvert the President’s veto power by adjourning while bills are on the President’s desk. If nothing else, Congress adjourns at the end of its two-year term, and before the Twentieth Amendment, the House of Representatives ceased to exist between March and December. There was no Speaker, no agent, no sworn members, no nothing. The President could be sitting with a bill on his desk, that he hates, and nobody to whom to return a veto. When Congress eventually reconvened, they could say, “Well, you had your ten days, and you didn’t do anything (because you couldn’t!), so that bill you hated is now law.” Given this power, Congress could play further games by pulling a surprise adjournment before the end of its term, leaving the President in the same predicament.
The pocket veto clause prevents this, by changing the default from law-without-a-signature to no-law-without-a-signature when Congress adjourns.
But alas, this solves one problem by creating another. What form of adjournment triggers the change–any adjournment, or only an adjournment which prevents a veto from being returned? If you say “any adjournment”, you open yourself to logical absurdities. Except when the Senate is in the grip of an all-night filibuster, each house of Congress adjourns every night. If the ten-day time period to sign a bill elapses at 6:42 p.m., and Congress adjourns for the night at 6:00, can the President pocket veto the bill? If the tenth day is a holiday, so that Congress isn’t in session, can the President pocket veto the bill? The designation of an agent avoids these problems by making it clear that adjournment of whatever flavor isn’t preventing the return of a veto–which, again, is the only reason to have a pocket veto clause.