Sorry for not answering sooner. I'm traveling at the moment and only look in sporadically.
As I wrote HotK, it became clear that there were two loves in Denethor's life - Finduilas and Thorongil. Yes, he was jealous of Thorongil and resented the usurpation, but by Umbar, he was ready to acknowledge the king because he loved and trusted the man. What I didn't realize until quite late in the story was that Thorongil returned that respect and affection, but not the trust. It's probably only in hindsight that he grasps what his flight did to Denethor, and he is profoundly guilty. I left you alone for too long. To him, Denethor was a big brother, much as Boromir was to Faramir, the powerful, decisive, elder who never stopped being the younger's defender. It shocks him to see this person fall so far. Even as he knows how and why Denethor's flaws led to his downfall, he has no less love for his big brother. He knows Denethor well enough that, without anyone saying a word, he instantly grasps what Denethor has done, addressing him correctly as Captain-General.
Denethor knows this. It is a source of solace and of anger. If you love me why did you go? Can you really forgive me, or is this just pity? Hence his reply that he wishes to see if his faith in Thorongil was well placed - the path to redemption for both of them (the flight after Umbar, the suicide/murder attempt) lies is getting this right.