How often are successful international teams actually enjoyable to watch? I think you have to go back to the Brazil team of 2002 as the last time a winner of the Euros or WC could be described as exciting.
Teams that win tournaments these days are mostly just efficient. France in 2020 were arguably a good watch but a lot of that centred on Mbappe's breakthrough. IIRC they weren't convincing at all in the early stages, either.
The most important qualities of tournament winning teams seems to be a solid defensive and winning mentality.
Under Southgate England were definitely solid on the whole, but lacked the ruthlessness and winning mentality (still can't quite work out how we didn't go on to beat Italy in the Euro 202 final). I was quite positive about Carsley to begin with, but after the last week I'm not sure he's going to be able to instil a winning mentality.
That said, no one immediately springs to mind who would be able to do so.