> Would a C++ compiler that doesn't accept "new" and "delete" be C++

Believe it or not, yes it would. It would not be a conforming implementation of any ISO C++ dialect, but it would be recongizable as a C++ dialect and mutually intelligible. Someone who knows C++ would understand programs written in that dialect, and could write new ones in it.

Something which is not a conforming ISO C++ implementation probably has good company; most production compilers are likely not fully conforming. Just not in gaping ways like new and delete being absent.