I really don't see how it would win sympathy from the Democrats. Lieberman left the Dem Party, after all. Could it be because Joe's a Jew? Starting to wonder.
I'm sure that's at least part of it. What better way to distance yourself from what many perceive to be the racist antisemites? It will get him a little support, or perhaps not "support" so much as a little less hatred, but I think Trump's reasons have more to do with him being a well respected Democrat that is regarded as relatively reasonable for a Democrat. Sensible even.
I used to be able to say that I knew of one Democrat that I didn't detest and that was Lieberman. When he left the party I could no longer say that.
I always thought Lieberman was one of the better members of their group. *He* did not piss me off like most of them. He was like Sam Nunn in that regard. Not a bad person .... for a Democrat.
But that doesn't mean I think he would be good for that position .... or any position in the Trump cabinet.
I don't think he would be great for the position, but I can see where Trump might think he would be better for Trump's politics.