During the first three seasons whenever someone compared DSC to the Kelvin Films, it made me recoil. It pressed my buttons. I got into a pretty heated fight with someone over it on the Zoom Chats here, which is why I stopped hosting them and gave it to someone else, then left shortly thereafter. So we're clear, as much as I'm a fan of DSC, I'm very "in the middle" about the Kelvin Films and I don't really like NuKirk that much, to be honest.
But I have to call a spade a spade. The opening sequence of "Kobayashi Maru" really did remind me of the beginnings of Into Darkness and Beyond. It was fun watching it, but it was the first thing that came to mind.
What they did with the Kelvin Films was model the beginnings after the beginnings of the Bond Films where they have a little pre-credit mini-adventure before they get into what the film is really about. So "Kobayashi Maru" started with a similar sequence showing Burnham doing her normal thing that she's been doing the last five months, before this anomaly appears and Everything Changes.
In TOS you had several action sequences pre-credits. Two immediate example are:
1) "Friday's Child" when a landing party beams down to Campella, which is made up of Kirk, Spock, McCoy, and a Redshirt, they seem the Campellans, then they see the Klingons, the Redshirt exclaims, "It's the Klingons!" and then he dies.
2) "A Private Little War" where, once again, the landing party beams down, then Spock is shot, and has to be beamed up and taken to Sickbay. Cue opening credits.
And there are more examples. But those aren't teasers that tell a complete story. Those teasers led into the meat of what the episode was going to be, and isn't something that could exist independently of the main story. Whereas the openings of "Kobayashi Maru", Into Darkness, Beyond, and the Bond Films could all exist independently as short stories in and of themselves.
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Also, the reason I say Burnham's story is more akin to paralleling NuKirk than Kirk in TWOK is because NuKirk was an inexperienced Captain who gains experience. TWOK's Kirk is about someone who's turned 50 (forget what the Star Trek Chronology says, it's obvious from the story that Kirk's turning 50, from the way he's acting and the way it's written) and is pining for the old days of when he was a Captain. He's careless because he's out of practice and has to get back into the swing of things. NuKirk and Burnham, on the other hand, are or will be careless because they're inexperienced and have to gain that experience.
Burnham is obviously a very different character from NuKirk, and I like Burnham a lot more, but the arc and where they're at in their career stages is similar. Although, unlike NuKirk, Burnham actually spent 10 years in the lower ranks. So maybe a case can also be made that Burnham is like Kirk either at the start of TOS or just before the start of TOS, but she's not at a point in her life where she'd be like Kirk in TWOK.