The claim is heard in many forms. "Surrender means giving up effort and thought. The Lord does everything, so the person has nothing left to do or to weigh." Someone says "it is all His will" and stops preparing. Someone reads "abandon all dharmas" as leaving duties, and the Lord turning beings "as on a machine" as proof that a person is only a puppet.
Why the claim is easy to believe
Three phrases, taken alone, seem to support it. The Lord asks for all actions to be surrendered to Him (Gita 18.57). He says that He turns all beings as though on a machine (Gita 18.61). He says to abandon all dharmas and take refuge in Him alone (Gita 18.66). The claim also holds a true part: something really is handed over. The question is what, and the text answers it in three places.
Test one: the Lord asks for thought
विमृश्यैतदशेषेण यथेच्छसि तथा कुरु ॥ १८.६३ ॥
Thus I have declared to you knowledge more secret than the secret. Reflect on this fully, and then do as you wish.
After declaring the most secret teaching, the Lord says: reflect on it fully, and then do as you wish (Gita 18.63). The Lord asks Arjuna to reflect, and the choice is left to him. A teaching of passivity would not ask for reflection and would not leave a choice.
Test two: surrender comes with the steady intellect
अथ चेत्त्वमहङ्कारान्न श्रोष्यसि विनङ्क्ष्यसि ॥ १८.५८ ॥
With your mind fixed on Me, you will cross over all obstacles by My grace. But if, out of egoism, you do not listen, you will perish.
With the mind fixed on the Lord, obstacles are crossed by grace. If out of egoism Arjuna does not listen, he will perish (Gita 18.58). The ego in this verse is a closed ear and not effort. And the verse before it joins surrender to the yoga of the intellect: actions are given to the Lord in the mind while the understanding stays steady (Gita 18.57; see Buddhi Yoga). The surrender asked for is made by someone who is awake.
Test three: the Gita never tells anyone to sit down
When Arjuna sat down with his bow cast aside, the Lord's reply included "rise" (Gita 2.3; see Arjuna's Despondency). The perfected one is described as always performing all actions while taking refuge in Him (Gita 18.56; see Freedom Through Action). And Arjuna, at the end, says that he will act according to the Lord's word (Gita 18.73). Nowhere does refuge come together with stopping.
What the claim gets right
Something is given up. The right is to the action, never to its fruits (Gita 2.47; see Action Without Claiming the Fruit). One fights with all actions dedicated to the Lord, with the mind on the Self and free from fever (Gita 3.30). The claim over the outcome and the worry about it are handed over. Effort and thought are not.
The better reading
Surrender hands over what is not ours: the outcome, the thought "I alone", and the grief. It keeps what is ours: thought, choice and work. The machine verse says where the forces come from, and the very next verses ask the person to take refuge by choice. A puppet is not asked to choose.
A useful question to carry is this: "Am I handing over the worry, or the work?" When the answer is the worry, the surrender is the Gita's own.
Where to go next
The ten verses are taken in order in the detailed reading. The question of free choice is answered at length in the deeper reading. The hub is The Final Counsel, and how actions are offered is in Offering All Actions.