historyfool, I basically agree with you but want to comment on the role you overlooked--military (and related, piracy). I read the Wikipedia article on Clipper Ships and it offered very little insight into the whys and wherefores of the timing of their introduction; many points raised in these replies here could be usefully incorporated into its history section I suspect. But it did remark that clippers were used both by navies and by pirates.
However, just as the low capacity points to a constraining limit in the commercial sphere, so does the importance of running light with heavy rigging to a clipper's speed suggest why their marital role would be peripheral and limited despite the clear advantage of speed. The Wikipedia article also comments that clipper ships had short service lives; being lightly built they did not wear well. Whereas, in the age of sail after the battle of the defense of England against the Spanish Armada, when cannon clearly tipped the balance away from the ancient dominance of boarding actions as a major, if not quite the only, mode of combat at sea, what mattered was being able to bring a devastating mass and rate of heavy, ship-killing firepower to bear on a foe. And although an advantage in weaponry might enable one ship to threaten to pepper another at such a long range the latter's inferior guns cannot threaten the attacker at all (the English in the Armada fight enjoyed just such an advantage, with superior guns) what was found to be the case, starting already with the Armada battle was that standing off that far, far enough to be perfectly safe, would do little damage to the enemy and use up all the powder and shot in short order. In order to inflict really punishing, crippling, ship-killing damage on the target, it was necessary to be willing and able to come into range of even an inferiorly armed ship's own weapons--a disparity in gun quality might mean the ship with the advantage suffers relatively little while the other is pounded into splinters. But the attacker will not get out unscathed!
Thus, however obvious it is that a British ship of the line of the Napoleonic War era was much evolved beyond 16th century galleons, both shared the characteristic of being quite strong, robust structures. A ship built on clipper lines with such a strong structure would have little room for provisions, much less extensive ammunition. It seems from comments above that new technology in basic ship construction had to be developed to allow the light hulls of the clippers, and apparently these new techniques were taken in the direction of doing more with less, rather than doing a lot more in terms of strength with the same masses as were used before. Thus clippers could handle the considerable stresses of their mode of operation--marginally. Taking shots at it from a set of cannon, though, one supposes the thin, carefully approached margins would be broken up completely in very short order.
Now with speed comes maneuverability, I would suppose--rather, at a high speed generally it is harder to steer, but the same heavy sail on a light sharp hull that can give speed can also give superior maneuvering at lower speeds. A clipper with a decent broadside (perhaps with not a lot of ammo to feed it for a long slugfest it could hardly survive anyway, but enough to give a punishing blast to a lightly armed merchantman) might well be able to outsail and threaten such a merchant ship caught out alone--thus the role of commerce raider and pirate is open to it, using high speed to evade naval forces that might come belatedly to the prey's aid or to avenge it. In a large navy, they could also serve as scouts and couriers, and perhaps even deliver crucial supplies. But clearly in a navy, the main battle line would have to consist of heavier, more conventional hulls that could slug it out with an enemy line.
A key feature defining a clipper, as set forth in the Wikipedia article, is that in addition to a light, water-slicing hull and a mass of sail aloft, is that the clipper keeps the sails up even in bad weather. How the light hull handles strong wind forces on its masts is partially a matter of technology, I suppose, but also (though the article fails to mention this and I might be mistaken somehow) seamanship. I don't know if a clipper crew was unusually large compared to a more conventional sailing ship of similar tonnage, but I would expect it must have been, and they would be constantly busy adjusting the rigging to take best advantage of the wind while avoiding overstressing the ship too much.
Thus, a clipper that tangles with other ships violently runs the additional risks that a defender can target the sails and rigging as well as the hull, and that vitally needed crew might be killed or otherwise taken out of action in a fight.
Something the article does mention is that clippers, tacking in powerful winds, often heeled over so much their leeward rails neared the waterline; it would be tricky and risky installing gunports to make for a broadside in hulls sailed in those conditions! I believe that watertight gun ports was something that had been developed in previous generations so it would not be impossible, but it would be another thing for a busy crew to keep an eye on. Along with securing the cannon against such drastic lists!
Thus perhaps it is more evident why, despite the obvious utility to a navy of faster communications, we don't hear a lot about naval clippers. Perhaps they existed but being relegated to an auxiliary role, never got the fame more robust if slower ships might earn.