Some fans - a lot of them to be honest - really hate Liverpool's guts for some reason. Some you understand why, but others are a bit of a mystery. Doesn't seem to be just the club either, but Liverpool people in general. If they don't, I have no clue why they think those kinds of "songs" are even relevant anymore, but they might want to save them for their home games - otherwise you're just riling up an entire stadium and getting them behind their team. Hell, Arteta managed that just squaring up to Klopp!
As for the game, good to see Rodger's tactics bite him in the arse. I suspect, though, they're just going to use this result as an excuse for the inevitable bumming they get off City, and then have another crack at us - they'll still get smashed though.
I know the Football League isn't exactly rolling in money, but the League Cup pays such a pittance in prize money I'm amazed PL clubs haven't just banded together to fuck it off. The extra games aren't worth it if it costs you players that can fuck up your season.