Personally, I don’t think it matters either way. When I was younger, I wouldn’t like it when players didn't sing it but I think it’s one of a plethora of things managers/players do outwardly that now I’ve realised sort of don’t actually matter and have no benefit/drawback.
On your second point, I don’t really think you can really pin Carsley into the ‘Southgate era’. He’s been in situ during his time, but Southgate didn’t employ him. If you look at their style of play and even the squad he’s picked, there’s a big difference between the two.