The excerpt suggests that Hamilton refused Lord Stirling's offer to be a military aide because he was "headstrong" and did not want to be subordinate to anyone. This implies that Hamilton was ambitious and preferred to lead rather than serve in a subordinate role. Therefore, the most accurate conclusion is that he wanted the excitement of the battlefield and the opportunity to lead, rather than serve under someone else.
So, the conclusion that best aligns with the provided information is that Hamilton refused Stirling's offer because he wanted the excitement of the battlefield.