Forcing the Papacy to recognize Imperial authority and getting rid of the Papal states are not the same thing.
True, but most likely the course of the successful struggle between the Staufen and the theocratic Popes - Italian Communes coalition sees the former affirming Imperial control over Italy, including Rome and Latium. They are not going to tolerate any independent centers of power in Italy, most definitely including any Papal states. The Pope is reduced to a position very similar to the one of the Bizantine Patriarch, they hold a lot of prestige and influence, but are allowed NO secular power base.
It is quite possible that the course of the struggle sees an early Great Schism emerge in Europe, the Empire holding Germany and Italy to the obedience of the pro-Imperial Antipope, while the theocratic Popes make an alliance of convenience with the King of France and/or England. Then it boils down to whom wins the war in Western Europe. Of course, unless the pro-Papal King wins an early crushing victory (and if France sides with the Pope against the Empire, most likely England takes the opposite side, and vice versa), which affirms the cause of the theocratic Papacy, either the Empire wins, then the Papal cause is crushed, or a stalemate develops that shall make the Roman Pope effectively as much a puppet of France/England as the Imperial Pope is of the Staufens. The most likely outcome of the stalemate is that the various monarchs can see beyond their differences and agree on their common interest to affirm secular supremacy and on some kind of compromise making the Council supreme in the Church.
Even if the Staufen are able to impose Imperial authority, the Pope will still hold sway over the city of Rome and the surrounding area. He will simply do so with the say-so of the Emperor.
No. Given past experiences, they shall see it is unwise to leave the Papacy any kind of independent secualr power base, and their goal in this struggle is to affirm centralized power base in both Germany and Italy. this leaves no place for a Papal state.
This arrangement won't last forever. The Staufen will eventually lose authority in Italy,
And what is ever going to cause this loss of authority ? If the Staufens are able to establish a solid centralized rule in Germany and Italy at this point, and abort the development of the Communes as an independent power structure, you shall have a big centralized bi-national state running from the Baltics to Sicily. It might break up if they undergo some terrible catasthrophe (a total military defeat against France, successful Mongol invaison of Europe, a long dynastic civil war, although the latter is much less likely with the uprooting of feudal decentralization) but is not a given by any means. At this point, the Emperors would have already wiped out all the most important centrifugal structures, the feudal princes in Germany, the Communes in Italy , and the theocratic Church. The most likely outcome at this point is that it evolves into a big centralized Empire much akin to a Western Russia all the way into the Modern age.
Mind it, this is an age where national identities are still very fluid and largely driven in their development by the political borders.