Now is, "Do this but don't choose to do this," likewise odd, so that we "see" some connection between choice and action?
The answer to this question is a very emphatic no. Both athletic psychologists and Zen practitioners believe in a state of what Zen calls no-mind or mushin. Simply put, one acts not from deliberative conscious thought, but rather from what might be viewed as habit, instinct, or spontaneity. In the martial arts, if one has to will a blow to land on an opponent, one is likely to allow the opponent to land the blow first. But one needn't use combat for examples. Normally when driving stick, a driver manipulates the clutch, brake, accelerator, and gear shifter without willing the individual actions, but rather willing the outcome. That is to say, a driver chooses to pass a car, avoid a pedestrian, recover from a fishtail on ice, not engage the clutch, switch from 1st to 2nd and so on. The same can be applied to professional sports performance, acting, and host of activities and is related to the psychology of flow which is an altered state of consciousness.
In psychology, action that is devoid of choice is known as compulsion and is mirrored if one considers thoughts a behavior by obsessions, and the DSM-V recognizes two personality disorders based a spectrum of obsessive-compulsive behavior, OCD and OCPD. Other exotic diagnoses exist including Tourette syndrome and alien hand syndrome. Thus, philosophy of mind has been validated by the scientific method in regards to choice and action being decoupled. Both The philosophical and psychological recognition that not all human behavior is consciously chosen goes back a long way into literature and predates the Western Canon of philosophy.
But how do philosophers cope with this more broadly? They tackle it under the concept of intention and intentionality, two related terms that mean very different things. The first means roughly a conscious desire to act to achieve a goal, and the latter is used by philosopher of minds to talk about how the mind is able to be "about things", how the mind represents or reflects the state of affairs in the world. Many people are familiar with this philosophical distinction through legal philosophy and mens rea, where a criminal has acted, but it is a question of whether it was a choice. Without the conscious choice, there arises an affirmative defense.
Ultimately, however, most people have had the experience of learning and forgetting skills. Learning to ride a bicycle can be a difficult enterprise, but once it is internalized, the coordinated actions required to enter a state of successful use are handled subconsciously, by what Searle simply the Background. If you fluently drive manual and suddenly looked at your feet when the light turned green and realized you have no idea how to operate the pedals, you've experienced the disconnect between choice and action.
