In which contrib file/function should a class_alias() statement be placed so that it is always processed, regardless of the url being served?
So is there a better place than my_module.module where I can put the
class_alias( ) statements, such that they are always processed?

Put class_alias() in the class file (after the use statements before class {} or at the end of the class file).
*.module files are not loaded by the class loader, they are included later in the kernel bootstrap process.