How much information does it take to meet the threshold of "sufficient"?
If the answer is "all" then sorry, that's just not gonna happen very often if ever - similar to real life, that way - rendering this definition of agency as meaningless.
If the answer is "some" then that represents a sliding scale between 1% and 99%; and at 1% the answer might as well be "any at all".
My position is that you still have agency even if the answer is "none", provided you're still free to make a (perhaps random) choice rather than have that choice made for you.
Unless I'm playing a fool as my character (which I've done many a time

) I'd hold the bolded as true no matter what system or playstyle I'm in; as that's realistically what any halfway wise character would try to do in the fiction, and thus to be true to the character that's what I'd have it do as its player.
And a system or GM that doesn't let me (try to) do these things is denying me my agency as a player to declare my character's actions.