It's impossible to speak for every Medieval writer or thinker, and opinions would have differed at the time just as they did today, but by and large I am not convinced by the idea that the First Crusade is best seen as a reaction to continued fears about an encroaching Muslim threat. The bulk of the Islamic conquest was centuries old by the 11th Century, so the immediacy of the threat you would have seen during the Merovingian period (when Muslim forces were still actively pushing out of Iberia) was largely a thing of the past. There was something of a catalysing threat in the form of the Battle of Manzikert (1071), which was a disastrous defeat for the Byzantines, but that was over 20 years old by the time of the Council of Clermont's call for Crusade. To be fair, Gregory VII had tried to launch some form of military expedition soon after the battle, so the idea of sending a military force east had been suggested, but it never came to be and was quickly lost amidst the ongoing Investiture Contest between the papacy and the Holy Roman Empire.

I don't think we can say that Western Christians weren't concerned about Islam and the control of Iberia and the Holy Land by Muslim leaders, but I don't think you can make a reasonable case that it was an overriding concern for them. If you'll excuse a modern metaphor, I think a lot of people in the 'western world' are probably concerned about extremist movements in Africa, but I think they're probably a lot more concerned about jobs, health care, and the politics of their own country and region. The 11th century saw some major conflicts within Europe (the aforementioned Investiture Contest, which spurred several civil wars in Germany, as well as the Norman Conquest, which followed on a lot of infighting within Normandy while William established control of his duchy, and many other smaller conflicts). The call for the First Crusade can actually be seen as something of an outgrowth of this. Urban II wasn't the only person claiming to be pope in Europe, and in fact he was living in exile in France because the German Emperor had declared his own pope and had control of Rome and much of Northern Italy. While not the only reason for his actions, Urban's call for the Crusade at the Council of Clermont is a defining moment in his eventual success at asserting his right as the sole pope in Europe, and relegating his rival to the ranks of anti-pope in the annals of history. Maybe you still get the First Crusade even if Urban II was the sole claimant to the papacy, but I'm not entirely sure you do. Obviously there are a lot of other factors at play, and Alexios' request for help is a big part, but I don't think an immediate and extreme fear of the threat of Islam was major one.

It's worth also noting that in his work The Race for Paradise, Paul Cobb argues that from the perspective of many Islamic groups it was Christianity who was providing the active and immediate threat, not the Muslims. While we focus a lot on Byzantine losses during the second half of the 11th century, it's worth bearing in mind that this was the only place where Islam was gaining territory. At the same time as Manzikert and its following losses Christian Normans conquered Sicily, territory that Islam never won back, and Christian princes were making increasing progress in Iberia. Some of this Iberian land was lost again (there was a lot of back and forth during this period), but the trend was not encouraging for Muslim rulers. While the perception of threat doesn't need to line up with actual threat, Western Christians could easily feel threatened even as they were 'winning', I think it's a harder sell. As a further note, in the 1080s many of the same Normans who conquered Sicily launched an invasion of Byzantium with the hope of carving out territory for themselves (one later participant in the First Crusade, Bohemond, actually surrendered his inheritance in Italy to secure lands in Byzantine territory). A full on invasion of Byzantium is not all that compatible with a culture that's terrified that Christian territory might be irrevocably lost to the hands of 'evil Muslims'.

Thomas Asbridge's The Crusades is probably my favourite general history of the subject, but I've heard great things about Christopher Tyerman's God's War as well.

For non-western perspectives on the First Crusade I like Peter Frankopan's A Call from the East, which is from the Byzantine perspective, and Paul Cobb's The Race for Paradise, which is a full history of the Crusades from the perspective of Islam.