I think this is generally a hard question to answer. I think, overall, it mostly depends on how much power the Pope had at the time of calling the crusade. Around the turn of the second millennium, there was a movement within the church to institute various reforms, such as removing clerical marriage, ending simony, and ending lay investiture. Nicholas II, Alexander II, and especially Peter Damian and Gregory VII are well known members of this reform movement. It reached it's first head during the Investiture Controversy between Gregory VII and Henry IV of the HRE with the latter's walk to Canossa in 1077. The controversy ended with the Concordat of Worms in 1122 between Calixtus II and Henry V that allowed the lay rulers to choose clerical replacements, but to not invest them with the ring and staff. This was a particularly empowering moment for the papacy and helped demonstrate what is now known as Papal Supremacy.

However, throughout the whole reform movement, the popes tried to enhance their own power. The First Crusade, called by Urban II in 1095, helps demonstrate how powerful the pope had become as they were particularly weak during the Ottonian Dynasty (919-1024) of the HRE. The papacy remained strong through the 12th century and Gregory VIII was even able to call a crusade that had all the most powerful leaders of Western Europe join (Third Crusade 1189).

A particularly powerful pope that comes soon is Innocent III (1189-1216). He called for successful crusades in Iberia and and against the Cathars in south France. He called the Fourth Lateran Council that helped develop canon law, the doctrine of papal supremacy, and ideas like lay responsibility (it also helped lay the groundwork of the Inquisition).

He is also notable for the Fourth Crusade (1202). This looked to be a very successful endeavor with the French very much behind it and the Venetians providing navigation; however, the Venetians were not sufficiently paid for their ships. This led to the crusade being shanghaied by the Venetians who first attacked Zara, a city sort of in their sphere of influence, and later Constantinople (sacked by Latin forces in 1204). Innocent wanted to capture Jerusalem, obviously, but he was not influential enough to overcome the commercial interests of the Venetians. Innocent was initially upset and later ambivalent towards the Crusade for its divergence, but was finally reluctantly accepting because claiming Constantinople would allow for the Latin Church to extend influence once again over the Greek Church. Overall though, the Fourth Crusade shows a decline in the traditional Crusader spirit, as the Venetians acted more out of their own interests than the interests of the Church. This is very different than what occurred during the First, Second, and Third Crusades.

Following Innocent III, we have Honorious III and Gregory IX. They were still in the vein of the "reformer popes" going back to Gregory VII. Gregory IX published the Decretals and instituted the Inquisition to go after "heretics", like Cathars, Waldnesians, Beguines, etc, which both helped increase papal power again.

During Gregory IX, there is also the reign of Frederick II in the HRE. He fought very hard against papal supremacy and was a distinct enemy of Gregory IX. He made a crusader vow, but didn't go on the Fifth Crusade (which failed horribly; 1213), so he was excommunicated. Then, he went on the Sixth Crusade, while still excommunicated, and actually won back Jerusalem and other lands in the Holy Land, which allowed him to claim the title of King of Jerusalem while he was excommunicated from the Church. This was a huge issue for Gregory IX and the Church as a whole for obvious reasons. Frederick's conflict with the Church continued with Innocent IV, until the former's death, which lead to the Great Interregnum for the HRE (1250-1273).

With the HRE now weak, the papacy remained one of, if not the, strongest actor in Europe. This is best reflected by Boniface VIII's Unam Sanctum in 1302 on papal supremacy (he was in conflict with Philip the Fair of France at the time). However, traditional crusader spirit was most gone as can be seen by the detour of the Fourth Crusade, the embarrassment of the Fifth Crusade, and the odd victory of the Sixth Crusade.

Papal power began to recede though as the papacy moved to Avignon and went through the Western Schism (1309-1417). This led to the rise of conciliarism (Councils of Pisa, Constance, and Basil 1409, 14-8, and 31-49) that saw the power of the papacy significantly weakened (conciliarism was denounced in the Fifth Lateran Council in the 16th century).

Around the same time, the French and English crowns were getting more powerful, the Reconquista was still successful in Spain (leading to power Castillian and Aragonese kingdoms), the Hohenstaufens came to the head of the HRE, helping to re-empower it, and city-states in Italy, like Venice and Florence, were becoming even more wealthy and powerful.

Thus, by the fifteenth century, the crusading spirit of the First to Third crusades was already mostly dead, the papacy was significantly weakened, and secular rulers were able to re-assert their authority compared to the popes.

Did Europe care about the final Crusades? In the same way as they cared about the first ones? No, probably not. Sigismund fought the Hussite Wars because he didn't want Bohemia to leave his realm of influence in the HRE. The papacy would continue the weaken as, for example, the Inquisition was taken over by secular kingdoms (like Spain and Portugal) as time went on. The final nail in the coffin for Papal secular power in Europe was the Peace of Westphalia in 1648.

Additionally, the rise of the Ottomans made Crusading less appealing because they were particularly strong; the discovery of the New World provided for new ways to spread the faith; and the "fall" of the Holy Roman Emperor (no longer viewed as above the other kings of Europe) made it difficult to organize crusades amongst all the actors in Europe.

In all, Europe probably didn't care much about the final Crusades as they did for the first three.

Sources:

The Medieval Papacy by Brett Edward Whalen

Power and the Holy in the Age of the Investiture Controversy by Maurren C. Miller

Readings in Medieval History by Patrick J Geary