Completing Shadowbringers and patches is just a matter of having access to Endwalker's MSQ in the first place.

What I think it means is that, at very least, your understanding of what is happening in parts of the storyline may be enhanced by being familiar with certain other optional story arcs, and it may affect the dialogue you get from other characters and even your own dialogue prompts at key moments.

Compare Shadowbringers and its relationship to other story arcs.

Most pressingly, the Crystal Tower, which was optional at the time. Familiarity with the story gave you a good heads-up as to what might be the truth of a major character, recognise him at a key moment, and basically unlocks the "best ending" by giving you an extra dialogue choice after the final battle.

But also lurking in the background, familiarity with the Alexander and Omega storylines will let you catch references, particularly in the Twinning, to understand how it all played out. You hear the music and you recognise it. The final boss of the Twinning isn't just some weird machine thing.

When the story writer for Endwalker was asked at Fan Fest for similar recommendations this time around, she said there wasn't anything quite as important as the Crystal Tower but that basically every side story in the game would have some significance, and particularly emphasised the Binding Coil and Omega.

Yoshida repeated these in another interview and also added Alexander to the list, though it may have been generically listing "important side stories" over Endwalker particularly, and Alexander is key to Shadowbringers.

So if you've done those three optional stories, you are probably all set. If not, then I would certainly do Coil and Omega at least, though I have no idea what the queue would be like. Coil is best done undersized through PF anyway.