The Klingons are obviously beside themselves and decide to take revenge.
I think you're misunderstanding what T'Kuvma, the leader of the Klingon sect, is doing here.
The Klingons, at this point, are deeply fragmented as a people. Something that Sarek notes later on
SAREK: An odd proposition given that their Empire has been in disarray for generations.
T'Kuvma is deliberately trying to start an incident with the Federation. T'Kuvma tells us this in the opening moments of the episode (transcript)
T'KUVMA: They are coming. Atom by atom they will coil around us and take all that we are. There is only one way to confront this threat. By reuniting the twenty four warring houses of our empire. We have forgotten the unforgettable, the last to unify our tribes, Kahless. Together under one creed, remain Klingon. That is why we light our beacon this day. To assemble our people. To lock arms against those fatal greeting is...(in English) We come in peace.
After the Klingon Burnham kills is brought back, there's this
T'KUVMA: Witness our brother, our Torchbearer, killed by the Federation interloper on our sacred beacon. I see you, as you see the end. Our Torchbearer honours us. First to die in our crusade for self-preservation.
They're most decidedly not beside themselves. In fact, T'Kuvma is elated that he now has a better reason to light the Beacon of Kahless and summon the Klingons to war. There's some trepidation amongst the Klingons over this
T'KUVMA: The honour falls to you to be our new Torchbearer.
OR'EQ: With respect, I find great wisdom in your teachings.
T'KUVMA: But?
OR'EQ: Can you be sure our brothers and sisters will answer your call?
T'KUVMA: They will because the prophecy commands all Klingons must
come to the light of Kahless when it shines in the night sky.
OR'EQ: Will ships really fly across the galaxy because of a fable?
T'KUVMA: So, you doubt? You dishonour only yourself.
The irony here is what plays out next gives T'Kuvma exactly what he wants, albeit not in the way he necessarily wanted. Burnham talks with Sarek and learns how the Vulcans were able to open dialog with them: by shooting first
BURNHAM: 240 years ago, near H'Atoria, a Vulcan ship crossed into Klingon space. The Klingons attacked immediately. They destroyed the vessel. Vulcans don't make the same mistake twice. From then on, until formal relations were established, whenever the Vulcans crossed paths with Klingons, the Vulcans fired first. They said hello in a language the Klingons understood. Violence brought respect. Respect brought peace. Captain, we have to give the Klingons a Vulcan hello.
GEORGIOU: If their intention is to attack, balling up our fists won't dissuade them.
BURNHAM: It would be logical for you to take into account my success rate during our seven years together, and execute my plan without further challenge before we're dragged into war.
GEORGIOU: Starfleet doesn't fire first. That's all, Number One.
Battle of the Binary Stars tells us how this plays out. Burnham, having failed to convince Georgiou to shoot first, knocks her out and tries to launch a torpedo in a "Vulcan hello". Georgiou recovers and stop her. Georgiou then does the very thing you think wasn't tried: she tries to de-escalate and, instead, projects weakness to the Klingons, which leads them to start the Klingon War
GEORGIOU [hologram]: We are on the outermost borders of Federation space, but make no mistake, you and your artefact are in our territory. We regret the situation has resulted in the death of your warrior. We offer you two choices. Leave immediately or open a dialogue with us. Hopefully, it is the latter, so that we can reengage with the Klingon Empire and prove to you that now, as always
T'KUVMA: Here it comes. Their lie.
GEORGIOU [hologram]: We come in peace. (hologram ends)
T'KUVMA: No! They do not! The come to destroy our individuality. Shall we rise up together and give them the fight they deserve?
KLINGONS: Remain Klingon!
T'KUVMA: Fire!
(The Klingon fleet opens fire.)
The TL;DR here is that T'Kuvma is kinda like a Klingon version of a Pope. He has a lot of respect, but no real political power. Since the Klingons respect strength, he believes they will respond to a war with the Federation, especially if they see the Federation project weakness, which they do. No negotiation would have worked without attacking the Klingons first. Had they disabled T'Kuvma's ship and then talked, things likely would have gone differently, since the Klingons would still be fragmented and unwilling to attack an ostensibly stronger Federation.
