In your scenario you are one guy alone getting attacked and defending yourself. Kirk on the other hand is an agent of an organization which is engaged in a three-player Cold War and committed to protect trillions of people. The well-being of his ship or the desire to crush the enemy is irrelevant in comparison to the political complications he thinks about ("you show them compassion - it may be the only way to earn peace with Romulus"). Sure, the calculation is wrong but he does not have all the information or does not totally understand the situation. If you do not know where the Romulan ship came from prisoners could provide intelligence or become an asset in negotiations so it is an option which has to be considered.
I am not saying that Kirk should have done anything to actually save the Romulan crew unless he was totally sure that the ship is no danger anymore which sounds practically impossible ... but his trigger-happy "let's kick a dead man because the Vulcan establishment dude allowed me to do it" after he has already shown that he is able to think about the larger picture and other options is strange.