1-We do have machinery that performs a resolution of next action/thought/belief (based on external input, current state, etc.) and we possibly just sense that action as "choice"
2-It seems pretty unlikely that the resolution process could be cleanly deterministic and repeatable given quantum stuff, meaning some step in the resolution process hovering on the edge of two options could go either way depending on the brain's electromagnetic soup's current state.
3-Our emotions are critical for our survival, driving choices that help us short and long term, and it seems that one of those core drivers is a sense of agency. Even if we don't have free will in the magic sense of causing electrons (or whatever) to suddenly turn left to support our choice, it would still be critical for us to have a sense that our actions and efforts all matter, so those feelings are probably built into the system.
Note: This assumes there is no special magic somewhere that defies our current understanding of the universe and it just hasn't been figured out yet.