Grain of salt since I can’t remember where I heard this, but my understanding is that TOS-ENT and the first ten movies is generally a single combined license, but all of the new shows need to be licensed individually, I’m not sure if the Abrams films are licensed as a package or individually, but they’re separate from the other films.
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Fun fact: The TNG episode of the same name was originally intended to star Patrick McGoohan as Ira Graves.
Not only do you not need to catch up with Disco, but the end of season 2 of Disco doesn't dovetail to the beginning of Strange New Worlds super-well when you watch them back-to-back; if anything, they feel like they lightly contradict each other.
Though if I were to recommend a single episode it would be "Through the Valley of Shadows" from season 2, which is probably the only "important" episode as far as Strange New Worlds goes, though the key stuff gets recapped in SNW.
The Short Treks episode Q&A gets some references in the series too, though if you don't watch it you likely won't realize you missed anything.
I'd probably just start the PIC era in 2399. We only see the 2380's in a handful of flashbacks. Prodigy definitely has some PIC connections, but it's far more a continuation of Voyager, and it starts like a year after season 5 of Lower Decks.