The "Landshut" hostage crisis marks the culmination of the German Autumn, which itself adds into the larger themes of German post-war society moving on from and dealing with its Nazi heritage that permeate many aspects of the film.
While the original was merely set in 1977 due it being made there, the remake decidedly draws a lot more capital from that setting by heavily emphasizing its political situation and integrating that into the storyline, beginning with moving the story from the south-western province right into the middle of Berlin, if not even directly at the wall.
As you noticed, the film really is littered with references to the German Autumn, which the "Landshut" plane hijacking is really just a part of. From featuring the "Terror" cover of the SPIEGEL at various points, to street riots, RAF bombings and references to the Schleyer kidnapping (himself a former SS figure). The "Landshut" taking might be the most prominent in the news at the point of the story, but the RAF terror is generally omnipresent throughout the entire film. And yes, this could first and foremost be taken as enriching the setting with a temporal and local colour, as you posit yourself already.
But not only that, it directly adds to the larger themes of German history and the society dealing with it prevalent in the film when connecting those incidents back to World War II and the Nazi regime from which they ultimately stem. With the Red Army Faction (RAF) being a left-wing extremist group and to some degree the radical successor of the '68 movement, a large part of what they were revolting against was the idea of the previous generation, that was ultimately a participant or even perpetrator in the Nazi regime, still having much of their control over the government and its institutions. This is a primary conflict in post-war German society and culminated in things like the German incarnation of the '68 movement and the RAF actions (the film at one point directly points out that one of the police inspectors was already a police officer in WWII times, where he apparently helped Dr. Klemperer in his search for his wife).
The general idea of society moving on after WWII while dealing with its responsibility is a major motif of the film and by extension the chaos of the German Autumn figures right into that. We strongly have this in the personal story of Dr. Klemperer and his wife. But you see this also in repeated mentions of how the dancing group is itself very proud of having survived through the troubles of Nazi times and WWII, not the least by virtue of Madame Blanc:
And she's tough. She kept the company alive through the war. Think about that, when the Reich just wanted women to shut off their minds and keep their uteruses open, there was Blanc.
The witch coven in the dancing academy does seem to want to stay largely apolitical to the more direct conflicts of society, which becomes clear when they discuss Patricia's cause, who was herself much more personally drawn towards the RAF's ideals:
Patricia was unwilling. Blanc believes that is the key...We should not have forced it. What a fool. What we offered her! She wanted to blow up department stores instead.
It seems like all they really want to do is stand their ground in an evolving society and keep the dancing company going. But there can be somewhat of a more politcal angle to all this if we employ the classic notion of understanding witches as a symbol for women revolting against a patriarchal society. Some of the dialogue mentioned above about Blanc standing up against the mother ideal of the Nazis can be taken directly that way. But there's other pieces of dialogue where the witches directly blame Dr. Klemperer for not listening to his wife or women in general. In fact it seems to go as far as posing him as a representation of the guilty men who ultimately caused the war and perpetuated the Nazi atrocities (from which he personally is spoken free by Susie in the end). During the ritual he repeatedly exclaims:
I'm not guilty! I remember everything. I am innocent! If there are guilty men in Berlin? Everywhere! But I am not one of them.
So what we could say is that, while not necessarily having any ties to the RAF and their goals, the witch coven might very well be fighting their own fight against society, a fight they are keeping on from WWII times, similar to how the RAF and its associates thought they were supposedly keeping up the fight against the Nazis. The witches and the dance company aren't just trapped among a tumultous society, they are very much themselves part of the conflicts. Yet another angle could be related to the internal cleansing of old baggage that Susie, in fact a reincarnation of Mother Suspiriorum, performs on the witch coven.
And pretty much simultaneously with the internal fight in the witch coven culminating, the German Autumn also ends, with the "Landshut" crisis getting resolved and the head figures of the RAF commiting suicide in Stammheim prison. If we understand the RAF as a consequence of a supposedly incomplete denazification process and then set its ostensible end in relation to Susie's final visit to Dr. Klemperer, where she tells him the truth about his wife before cleansing his memory, we might see this as a general symbol for the German post-war society finally being able to move on from its Nazi heritage while accepting its responsibility.
Wir brauchen Schuld, Doktor, und Scham...aber nicht deine.
(We need guilt, doctor, and shame...but not your's.)
There are probably various layers to this (and I won't claim to have fully grasped the meaning of everything that happens yet), but in general we can say that the references to German Autumn aren't just enhancing the "worldbuilding", but directly add into a general post-war politcal thematic of the film, which is ultimately politicizing the very film and its story in a way that the original didn't intend to.