Whether or not Terry Pratchett intended GG to be the beginning of his best known series of Discworld novels, it seems fairly clear to me that by the time he wrote the end of it he knew it was going to be the first of a series—that he had found a setting and a group of characters with which he could write his comic satire. He could, and did bring in other characters, and these characters change and grow in the course of the series.
Part of the reason for some of the inconsistencies in these books is that, like life, they grew and developed. For instance, Carrot’s sword in this book is declared (presumably by Magrat Garlick) to be quite the most unmagical sword she’s ever seen. It’s also described at that time as “… a length of metal, more a sword than a saw but only just.” Yet in this book the sword seems to have something approaching magical status---when Vimes wields it to save the Patrician. And the Patrician examines “the rusty blade carefully,” then asks how long Vimes has had it. Carrot owns the sword as his and then Pratchett puts in a fascinating bit (the first time I think we’ve heard of these) about the trousers of time.
“Vimes felt the air thicken, as though history was clustering around this point, but for the life of him he couldn’t think why. This was one of those points where the Trousers of Time bifurcated themselves, and if you weren’t careful you’d go down the wrong leg -- “
Prachett interjects a short passage about Wonse’s encounter with death and then picks up the rest of the sentence. “—and the Patrician handed the sword to Carrot. "
In this passage, Prachett has swiftly and slyly pointed Ankh-Morpork down the leg of the Trousers of Time that sees Carrot as the rightful king of the city, but one who doesn’t want the responsibility. He will act if necessary, but he prefers to leave it to the Patrician. (Or more accurately--Pratchett doesn't want him to actually function as King.) And that sword, contrary to Magrat, is magical—but only in the hands of the right man or men.
After dismissing all the others, Vetinari has a revealing and important bit of advice which he gives to Captain Vimes. We see into Vetinari’s mind, to the way he runs his life and manages the city. It seems that Vetinari classes himself as one of the “bad people” , but as he says to Vimes "... you fellows really need us,” because bad people know how to plan. Good people (like Vimes) can be good at overthrowing the bad people, but they don’t have the knack for ruling the world.
What Pratchett has done in this passage as a whole is to set up the future relationship of these two men, a relationship which will grow and develop through the novels. Vetinari has, now, totally changed his opinion of Vimes, and sees that he is useful. Vimes is, and always will be. a tool not a ruler—but he has an important part to play in the future of Ankh-Morpork. And Pratchett will use Vimes far more than Vetinari to point out the foibles and abusridities of the human condition.
It is Vimes’s sudden awareness of this change in the nature of his world and his perception of it that brings forth the uncontrollable laughter. Prachett ends the chronicle of Vimes's laughter with “Laughed for the world and the saving of souls.” Vimes has seen his place in the world, has recognized the pretense and folly of most of the world, but also his role (as Vetinari’s terrier) in the saving of souls.