I haven't tried flutter yet myself but I've watched all their external presentations since they started the project and spent a lot of time on their mailing list.
The super short take-over-the-world plan for React has always been in 2 steps:
1) Abstract away the underlying imperative UI representation so that most of the code you write is React.
2) Swap the underlying representation with something designed for React specifically.
What Flutter attempts to do is 2) directly. Let's ditch the entire platform and recreate one from scratch. This is something that we've always wanted to do at some point in the future but we've always felt that it wasn't going to work because it was too much of a big-bang rewrite and there was no incremental adoption story.
I really want them to succeed as the model they are using is a much better one IMO than most of the current imperative-based platforms we have today.
As for our ambitions, I'm really happy that we're exploring it with React VR. This is essentially React with a backend that talks directly to OpenGL. This way we can remove all the middle mens and get absolutely awesome performance and developer productivity.