A worse Austrian defeat, leading to a military collapse and a harsher Peace of Prague is most likely going to start an event chain that causes the political collapse and partition of the Habsburg empire. However it is in all likelihood going to happen in a few years since the war (say within a decade) rather than immediately during the conflict. In 1866, conditions are not yet fully conductive: Germany has not formed yet, the serious obstacle of France has not been removed yet, and Prussia-Italy could not or would not afford a two-front war.
Picture this broad event sequence:
1866: Prussian victory at Sadowa, Italian victory at Custoza. Emboldened Prussia and Italy make maximum demands (all Irredentist claims for Italy; Saxony and Bohemia-Moravia for Prussia). Napoleon III makes pro-Austrian diplomatic pressures (backed by threat of military intervention), Prussia and Italy, unwilling to face a two-front war, accept a compromise. The Peace of Prague awards Prussia OTL gains, Saxony, Austrian Silesia, and northern-eastern Sudetenland, and gives Italy Veneto, Trento, and Gorizia-Gradisca. Prussia and Italy are satisfied with the outcome of the war, pissed off with France, and reaffirm their alliance. The peace allows Austria to avoid immediate political collapse, but the humiliation destabilizes the Empire. Prussia creates the North German Federation.
1867-70: growing tensions between Prussia-Italy and France lead to war, out of various possible issues (Luxemburg, Rome, the Spanish Succession) where compromise proves impossible. The Italo-Prussian alliance wins a decisive victory. Destabilization of Austria deepens: Austrian Germans experience a massive upsurge of Pan-German feelings (heightened by the Franco-Prussian/Italian War), the Hungarians make bold demans that make the Ausgleich compromise insufficient or impossible, the Czechs, Croats, and the remaining Italians get restive, too.
1871-74: The German Empire is created. defeated France is forced to make peace: the treaty of Frankfurt cedes Alsace-Lorraine to Germany; Nice, Savoy, and Corsica to Italy. France recognizes Tunisia in the Italian sphere of influence and pays heavy reparations. Further cessions of French colonies to Germany-Italy are possible. Internal tensions in the Austrian Empire reach the tipping point: the Hungarians secede, there is a Pan-German uprising in German Austria. The neighbor great powers organize the partition of the Habsburg lands: Germany gets German Austria, South Tyrol, Czechia, and Slovenia; Italy gets istria and Dalmatia; Russia gets Galicia and Bukovina; independent Hungary keeps the Lands of St. Stephen and a confederal union with Croatia-Slavonia.
In principle, I do agree with the timetable: with a better Italian performance in 1866, it is almost a given that the Italo-Prussian alliance will be continued.
I am quite sceptical on the real possibility of harsher Prussian demands at the peace table in Prague: first of all, Prussia has already gotten very significant land advantages out of the war (Hanover, Schleswig-Holstein and other minor territories annexed outright, and Saxony forced into the North German Confederation), and has to digest them without creating too many ripples. Taking Bohemia-Moravia (even if possible) might be more of a problem than an opportunity, at least in the immediate: there is no "German Empire" in 1866 and I would be at a loss to imagine a political frame where it could be put in; even annexing the Sudetenland might be too much. In second place it is important for Bismarck to get a peace signed as soon as possible, and while all other players are still stunned by the quickness of Prussian victory: Nappy has clearly misjudged the duration of the war and is just trying to muddle his way along (while it would have been in his interest to mobilise as soon as the declaration of war became public, if not even earlier than that - the Italo-Prussian diplomatic dance was not really an impenetrable secret: if one wants a proof that the guy was a wannabe this is it), Russia was friendly to Prussia (in particular after the support received during the Polish insurrection) but if the Austrians really implode, the czar might be tempted to take a bite and this would open a different can of worms. IMHO the benefits of a quick peace are much greater than the risks of going for some other not-so-significant annexations. Therefore I do believe that Bismarck made the right call for the particular moment. Btw, the short duration of the war works also against a sudden implosion of the Habsburg's domain: it would take time for the centrifugal forces in the empire to organise and act and, as far as the German nationalist component is concerned, they lack focus.
IMHO, a war between France and Prussia/NG Confederation is also a given, and cannot be delayed too long. In the 15 odd years since Nappy took the French crown the European situation has changed dramatically, and certainly not to the benefit of France. In Italy the Habsburgs have been expelled, but instead of a scenario where France could play a significant role as mentor and protector of mid-sized kingdoms a single unitary kingdom not only has been established, but has also proven that they could go forward and complete the unification process without French support and even against French desires. Even worse the Roman question is preventing him from gaining at least the benefit of a grateful Italy: Nappy's power base is among the catholics, so he cannot ditch the pope, but at the same time he cannot convince Pius IX to make a step back and compromise, much less convince Italy to renounce to Rome. In the Germanies the previous scenario featuring two major powers at odds with each other and a handful of other independent states has been wiped away, and replaced by a North German Confederation fully controlled by Prussia, with Austria removed as a player and the still independent southern German states cowed by the Prussian victory. Once again this has happened against his wishes and it is a monument to the failure of his diplomacy, in particular the repeated attempts to forge an alliance with Austria. His diplomatic space has also been restricted by the bad relations with Russia - the repression of the Polish insurrection was once again carried out notwithstanding the protests of France. He'll certainly try to purchase Luxembourg TTL too, but there is no reason why Prussia would wish to be more generous than OTL, nor there is any reason why the conference of Powers should be more sympathetic to France. The Roman question is not going away: if anything, Italy has come out significantly better from the 1866 war, so they will be more proactive toward this final acquisition. The Spanish succession is also an issue which cannot be postponed: ITTL the choice between a Hohenzollern or a Savoia on the throne of Madrid is not an acceptable one for France. In a few years Nappy's back will be against the wall: either he rolls up and shows his belly or he has to go for a war. My feeling is that 1867 and 1868 will be mostly taken up by the Luxembourg shenanigans, the Hungarian Ausgleich (or at least an attempt to patch up the relations between Austrians and Hungarians - but ITTL there is no guarantee this will work) and the diplomatic negotiations about the Spanish throne. By late 1868 Nappy will have lost the chance to buy Luxembourg, and will have to harden his position on the other issues (don't tell me he'll try and invade Belgium: if there were a remote possibility to make the French position worse this would be it). In the meantime he'll have tried to forge an alliance with Austria and Italy against the NCG: Italy will refuse, and most likely the same will be the Austrian answer (unless von Beust prevails against all odds, but the Hungarians would certainly be very unhappy). On the Prussian side, the peace treaties and reassurances with Bavaria etc. would be the same as OTL, and the Russian counterguarantee would be in place. The war has to come either in late spring 1869 or late spring 1870, and ITTL the spark might be an Italian occupation of Rome.
There is really no reason to anticipate a different outcome: while I do not expect major breakthrough on the Franco-Italian border, given the constraints of geography, at least France will have to keep some troops there. There is also no real reason to expect a faster French mobilization than the did in 1870. So the war will end up in a French defeat, the German empire will be proclaimed and Italy will get some (or all) of their claims.
The problem with this scenario is that now the German-speaking component of A-H has a lodestar, and the pan-German nationalism will surge. There can be different ways to play the end game, including some forays into Bosnia to create support for the Habsburg regime, but I would not bet my money on A-H surviving past 1875.