"I can't picture him wanting to talk about his friend's 'feelings'--what do others think?"

It didn't strike me so much as Sir John wanting to talk about feelings---it just seemed as if Sir John had fallen into Mrs. Jennings' habit of joshing everyone about their supposed love interests. And there was (for me) a touch of something else in Robert Hardy's performance, a bit of "my friend is despondent and I'll say something to encourage him." All those lines like, "And why not, I say--she would be a most fortunate young lady" seemed like a good-natured attempt to liven Brandon up a trifle. It doesn't work, of course. Still, I love the look of disbelief on Hardy's face in your picture, as if he can hardly believe that Brandon thinks of himself so "meanly."