He was not inviting a 'potential spy' to his quarters. He was inviting a man who was only under suspicion due to his heritage to his quarters. "Isn't it true you are a quarter Romulan, that it is a ROMULAN heritage you honor!" Like those are equivalent statements. Even accusing Simon Tarses is the same as accusing some random person on the street of being a terrorist just for being Muslim. I agree they took Admiral Satie over the top, but her point stopped having any validity the moment they caught the original spy and started randomly accusing anyone with even any slight circumstantial reason. It's shown time and time in history, when you get hysterical about enemy spies to the point that you start accusing anyone without any concrete evidence, you objectively and inescapably start doing more harm than the enemy spies possibly could do on their own. There were Russian spies in America during the Cold War, and the McCarthy hearings caught 0 spies and destroyed hundreds of lives. Satie was doing exactly the same and Picard was unquestionably right.