Maybe the term 'postmodern' is not the best, but it is describing a real development. Yes, postmodernity is modernity, but it's modernity without any faith in itself, which is an essential difference from early modernity. While many intellectuals did describe the chaos and sense of dislocation of modernity, there was also a great deal of optimism. We still have some optimists among us, but I think for the majority of people joining modernity now, their sense of a new and better order emerging is relatively brief. Postmodernity seems like a reasonable term for this.
The feeling of modernity, originally described by writers well over a century ago, is today shared by more people than ever.