While what you are saying makes a lot of sense, but it seemingly ignores the concerns of a people who, not too long before, had been made aware of the dangerous notion of Muslims having dominion over even an adjacent region to their own. I do know that the Papacy was gaining in power and popularity leading up to the Crusades. As such, and I believe what you say about getting rid of the king to be absolutely true, this is still lacking a component, that being a reason for the populace of Rome to stand behind their new "king".
"We are now expected to believe that the Crusades were an unwarranted act of aggression against a peaceful Muslim world. Hardly. The first call for a crusade occurred in 846 CE, when an Arab expedition to Sicily sailed up the Tiber and sacked St Peter's in Rome. A synod in France issued an appeal to Christian sovereigns to rally against 'the enemies of Christ,' and the pope, Leo IV, offered a heavenly reward to those who died fighting the Muslims. A century and a half and many battles later, in 1096, the Crusaders actually arrived in the Middle East. The Crusades were a late, limited, and unsuccessful imitation of the jihad - an attempt to recover by holy war what was lost by holy war. It failed, and it was not followed up." (Bernard Lewis, 2007 Irving Kristol Lecture, March 7)
Leo IV's actions to fortify after the sacking does show his concerns; with "Leonine City" with calls to invest into this as a means of defense from future incursions.
https://dispatch.richmond.edu/1860/12/29/4/93
A decent (Catholic bias) summary which you can find references for fairly easily: https://www.newadvent.org/cathen/09159a.htm
unfortunately it's hard to find this pdf without signing up or paying money but there are some useful figures if you scroll down
https://www.academia.edu/60028806/The_Surviving_Remains_of_t...
Showing the re-enforcement as well as a very clear and obvious purpose to it in light of when it was built.
I would recommend puttering about Lewis' work as well as the likes of Thomas Madden as well. If you are really adventurous you can dig up the likes of Henri Pirenne and his work on the topic; he argues that literate civilization continued in the West up until the arrival of Islam in the 7th century, Islam's blockade, through their piracy, in the Mediterranean being a core contributor in leaving the West in a state of poverty, and when you lose the ability to easily find food usually then literacy is placed on the back burner. Though that's just a tangent for another day, it's very interesting and he presents pretty decent evidence for his suppositions iirc.
Although if Pirenne is correct then the sacking of St. Peters carries a different tone, not one of just a "one off" oopsie but a sign of the intention of a troublesome and destructive new enemy setting their sites on Rome itself, not content to keep to the sea and to the East. It was a clear message to the people that they could be next in line (this is my opinion of course).
If you are American I would simply remind you that even now today you hear cries of a little nation across the sea being an "imminent threat to democracy" while our historic enemies are LITERALLY at our door just South of us and they have been there for several years now sitting in their little bases waiting for something. (I'm unclear as to when exactly it all started)
The notion that a Pope could give the people a reason, especially those who have felt the economic pressures, as well as the memory of a raid in their own home by the same aggressors, is possible. Being compelled to engage with an enemy that is a decent distance away is very believable.