In short then, the environment within which we find ourselves is causally determined; our options of choice represent a matrix of possibility that is limited. But the choice that is made itself is not made without our input.
But our input is causally determined.
For a "choice" to be made that is "free" then it surely needs to be selected without causation for the selection.
As for your last sentence - it's circular (or something like that) - "choice" surely requires the input of the one making the choice?
To me it is no more meaningful than saying "But the movie that is made itself is not made without a camera"... the latter half merely compounding the definition of the first part.
The question is... is there "choice".
There either is, or there is the illusion of choice.
To me for there to be real "choice" (as opposed to the illusion of) there needs to be an influence not caught up in the causal chain, but also that is in accordance to the "mind"... i.e. so not merely random.
I am not aware of any evidence of this.
And surely if an uncaused influence is in accordance to the "mind" there is likely to be a causal link?
It therefore seems more rational to me that "choice" is an illusion that comes after the event... even perhaps simultaneously... as the brain realises what it is doing and convinces you that you wanted to do it.
Sometimes it doesn't bother telling us, and we find ourselves doing things without thinking about it... without "choices" seemingly being made.
But the illusion is pervasive. And I would suggest (without backup/evidence/support) that it could even be a necessity for self-awareness.