The default splash screen cannot be overridden with Dart/Flutter alone. They are controls shown by the native Android/iOS context while the Flutter runtime is initializing. As such, any splash screen widget you create inside Flutter will show after the default splash screen. (Even if you completely remove the default splash image, the app would just be a blank while screen until Flutter finished loading, and that's bad design.)
There are instructions on how to change the default splash screen here. If you are well-versed in native Android or iOS development, you could take it a step further than just a simple image if you wanted.