As far as House Lannister is concerned, the Iron Throne belongs to House Baratheon; indeed, they must uphold that fiction at every point if they are to have any pretense of legitimacy. Hence, the debts of Robert Baratheon are passed on to his heirs, Joffrey and Tommen.
Now, it is entirely possible for Tywin to forgive those debts, as Tyrion asks him to. Tywin’s refusal is more personal in nature; he’s spent his entire life paying the king’s debts, first with Aerys II and then with Robert, and he views the reign of King Joffrey as where he will be repaid in every sense of the word.
Tywin’s refusal, and Cersei’s blind spot in this matter, points to a certain failure of imagination on the part of the Lannisters, that they don’t think of the Iron Throne and House Lannister as part of a single institution – in which case, the best strategy would be to forgive the Iron Throne’s debt, massively improving the Crown’s financial standing and allowing them to deal with Braavos and the Faith without need for default or concession – but rather see the Iron Throne as a means of extracting power and wealth for House Lannister.
What’s always surprised me isn’t so much Cersei’s idiocy in this matter, because Cersei isn’t that smart, but Kevan’s.
Cersei was very silly to trade the Faith Militant for debt forgiveness, but with that done, the Kevan Lannister Regency could have followed it up with the one-two punch of having House Lannister not only forgive the Crown’s debts, but handing a huge whack of gold over to the Iron Bank as well. Instead Kevan seems to be floundering around on this score… and Kevan is supposed to be one of the smart Lannisters.
Kevan is dutiful but not imaginative, I don’t think he would act against Tywin’s wishes.
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