Carsley does seem to have a bit more about his game than Southgate does, but I do fear a bit with England that they're not going to respond to another 'pull' type of manager. I was reading about this once that there are two types of manager, a pull manager and a push manager. The simplest way to put it was the 'pull' manager is very much about bringing everybody together and trying to play above the sum of your parts, a 'push' manager was more about challenging your players and placing expectation on them to deliver.
A pull manager would be more suited to a smaller team trying to punch above its weight, a push manager would be more suited to a bigger team expected to win trophies.
I think Southgate was very much a pull manager, which was what England did suit back in 2016, but have since become a team who requires the latter. I can see the logic in Carsley but I do think we've hit a point now where we need a manager who will demand trophies of himself and his players. Mourinho for all his style of football criticism I think has become suited to knockout football and it's no coincidence he's won Roma a European trophy, made the final of another (only to hit the most unstoppable of forces - Sevilla in the Europa League) yet probably underperformed in the league.
The more I think about who England should appoint the more I think Thomas Tuchel makes sense. I think there is scope for things to go wrong under Tuchel, but he does have a trick up his sleeve in cup competitions.