I agree it is partly his fault though the movie has the subtly of a sledgehammer and frames Keating positively. Its uncomplicated sentimental schlock.

the movie stacks the deck against us here by framing Keating as a rebel fighting against the crushing traditionalist forces of conformity at Welton. Case in point, the textbook scene, fun but manipulative. The textbook the boys are using is absurd. It has an introductory chapter about graphing the quality of poetry which Mr. Keating has them gleefully rip out. Of course we're going to root for him telling the boys to subversively rip out the dumb chapter. What if the scene was more realistic though? What if he had them rip out the imperialistic poetry of Kipling and instead study the beat poets? There is an argument to be made here, but its a messy and complicated one and Dead Poet Society strives to be simple. Its playing to a broad audience who want to feel like non-conformists and so it constantly presents you with a straw man instead.

Notice we never see Keating actually "teaching" in a traditional sense. He's more interested in influencing how the boys think, specifically that they think like him. Keating acts more like an Welton Oldboy (which he is) who went off on some grand adventure and wants to share his travels with the younger boys. He's that guy who comes back to teach at his old high school to relive his glory days and impressive impressionable teenagers (it ain't hard) rather than deal with his actual peers. However, he treats them like his university peers when arguably they're not ready for that. Picasso had to be trained classically, he needed to know what the rules were before he could break them. Keating clearly does not understand that. He received foundational knowledge about poetry and literature at Welton and then went to university where he learned how to break those rules. Now he is back to teaching to break the rules they have never learned because he SHOULD be teaching them.

I agree with OP, Keating gives an unnecessary and unrealistic sense of urgency to a bunch of high school rich kids. A more interesting movie would be willing to frame Neil's death as partly Keating's fault for manipulating the boys to think like him rather than actual teaching them to think for themselves. If you want to see a movie that actually does that well, see The Prime of Ms. Jean Brodie.