The caricatures were the straw that broke the camel's back. It's not like these people were like "OMG I love France and the West so much" one day, then saw the caricatures and immediately decided they were killing people. That's a very simplistic way to view things.
I'm sorry to say, but you're the person having a simplistic view in this scenario. You're arguing from ignorance, and conflating together separate problems. Again: there's no doubt that the general attitude of the West towards the Middle East can and does contribute to the rise of Islamist extremism, but in this case, it's completely irrelevant.
Please explain what happened then. My understanding was that a teacher was showing cartoons of Mohammad in class and the student decapitated them.
Prepare to have fun, because it's an adventure.
A teacher, Mr. Paty, during a literal class about free speech (which is why this is being tied to free speech so much), decided to bring up the Charlie Hebdo attacks (for obvious reasons). Since he was aware that this might be a sensitive subject, he warned the Muslim pupils in the class about what he was going to do and offered those not wanting to see the drawings to step out and take a small recess instead if needed. The class itself caused no issue, and none of the students attending it are further involved in the story in any way.
An unrelated pupil, a girl who did not even attend the class but had recently been excluded from another class entirely and was angry about it, heard about Mr. Paty's class. She then complained to her father, falsely portraying the class, a class she never attended, as having 'stigmatized Muslims'. Her father took offense on behalf of the non-offended pupils who actually attended the class, based on said false statement from his daughter, and doxxed Mr. Paty far and wide. He then proceeded to lead a social media smear campaign against Mr. Paty, complete with videos calling onto people to 'do what they must'.
The CRIF (an organization supposed to fight against Islamophobia, but which has a terrible track record with fact checking, leading to it being exploited by bad actors on the regular) happened onto the social media campaign against Mr. Paty. True to their track record, they did not check things in depth, and backed the father in his claims. The claims literally based on a teenage girl's lie.
A Charlie Hebdo attack survivor brought the issue up, pointing out that they were participating in getting an innocent targeted; rather than investigating further, the CRIF doubled down and launched a lawsuit against her for defamation.
Yet another person completely unrelated to the original events, an eighteen-year-old Chechen born in Moscow who just happened to live nearby and had fallen straight into identitarian Islamism less than six months earlier, fell on this whole shebang, overinflated by several more layers of lies, on his social media. Thanks to the father mentioned above, the workplace of Mr. Paty was known far and wide. The Chechen man went to wait there, bribed several students to tell him where Mr. Paty lived, and proceeded to assault and decapitate him in the street in broad daylight, then post it all on Twitter.
The CRIF had, meanwhile, continued to refuse to apologize for or recognize that it was participating in putting Mr. Paty at risk... and so found itself in the position of being in the middle of a lawsuit against a woman for telling them "You are going to get Mr. Paty killed" when Mr. Paty did, in fact, get killed. A rather large amount of people, exasperated with the CRIF's refusal to do its due diligence before taking action, began to loudly demand its dissolution.
(Far Right people of course jumped on the occasion as well. Any modern conflict with an internet base instantly becomes a magnet for bad actors.)
Yet more people, who had missed out on the sequence of events, arrived late to things and got for only information whatever tidbits came across their own social media grapevine. They saw people demanding the dissolution of a group purporting to fight Islamophobia, and decided that the demand was itself Islamophobic. Thus launching a counter protest and raising general exasperation even further.
Insert much facepalming by all the sane people involved, and the insane, muddled media shebang happening ever since.
I'm completely with you on the whole "backing religious extremist leaders has consequences" thing; I myself marched in the (successful) demonstrations which prevented France from joining the US debacle in Afghanistan and Iraq, and the (sadly less successful) ones against the intervention in Libya (which is its own adventure, and which France's current lawsuits against former president Sarkozy are a direct result of). The constant messing with the Middle East is a fucking disaster by any and all metrics. But Mr. Paty's death has about as much to do with US strikes in Afghanistan as it has to do with the fall of the USSR.
(Now if you want a modern event in France which is directly tied to generalized bullshitting and intervention in the Middle East, look no further than the band of about two hundred Gray Wolves-affiliated Turks who went on a bona fide pogrom in the streets of Lyon while clamoring for Armenian blood. That one was a nice little bit of complete disaster. Ironically also started by someone lying on social media.)
Honestly, at this point, it's precisely unchecked information and lies on social media that's causing most of the deaths. Hence my weariness at seeing you, among others, jumping on the occasion of an incident you don't know the details of to attach it to your own chosen cause. Please. Don't. There's definitely a place for the argument, but Mr. Paty's murder isn't it.