Boiardo died in Dec. 1494, so it was surely 22 before then - as early as 1460, although we have no idea when. In other respects, too, the probability of your supposition seems low. The Steele Sermon was delivered before it was put into the written form that survived. That the Fool was already a special card is indicated by its special placement outside the main list in the written version. I do not see how a change from 20 or 21 to 22 would likely lead to a change in the name of the game, unless the addition included the Fool. I can see how a change in the role of the Fool could lead to a change in name - but in Italy. In Bologna, Florence, and Piedmont, the Fool at some point acquired the ability to replace other cards in the scoring of combinations. Piscina writes (p. 15 at https://www.google.com/books/edition/Co ... frontcover; the Italian is on p. 14) :We don't know when or where the complete structure of 22 allegorical cards was consolidated. I think there's a possibility that this structure called Tarot was consolidated in France. I don't know if before 1495 there were decks with 22 allegorical cards or if the maximum was 20 or 21, as Rosenwald shows. It's likely that the name of this game didn't exist before 1495. In short, perhaps the French invented the name of the new variation and structure, or even the deck that includes the Matto (as a special card).
If it had just been Piedmont, and this was also a rule in France, I could see how the French could have led to that change. But Florence and Bologna count against that. And that rule did not get adopted in France, as far as known. Nor in Ferrara, at least by the time of Lollio.I will also say why the Fool is so much desired by Players because not only it is very good and useful, in the combinations it can take the place of the King and the Queen, meaning the great Lords and Princes, of Knights, meaning men of average condition, of the Page, interpreted as the rest of human generation if any of them is missing; this wants to represent that most of this sort of people has some kind of closeness and familiarity with the Fool, being the one that replaces them when they are absent.
About the dog on the Fool card playing a role in the name-change. We have to ask, when did that dog appear on the card? It is not on the Visconti-Sforza/PMB card, nor the Charles VI, nor d'Este card of 1483, nor the Sola-Busca, nor the Cary Sheet. It seems a later development. Once it was there, yes, I suppose, it could have suggested "chien" or even "talon" as the name of the small pile of cards discarded by the dealer. But that would have been later. Other games used the word "kitty" for their small piles. "Chien" could have been suggested by a French perception of the animal on the Fool card, which follows the dealer through the game.
Also, since throwing down cards is a characteristic of many card games, including ones existing before the game called "trionfi", (see Pratesi's essay "In the beginning was diritta"), it would not single out one in particular. Also, the cards discarded by the dealer are not used in trick-taking, so in that sense they are discarded cards but not cards discarded by players (players in the sense of those participating "in play", i.e. trick-taking).