Thanks. It seems that the born rule problem is a strong smoking-gun signal against the fundamentalness of semi-classical gravity. As for your objection drawing from the Bohr-Karmers-Slater theory, it seems that a manifestly, experimentally observable quantum phenomenon (like photoelectric effect) rules it out. But, are there such quantum gravitational phenomena which have been observed ? I don't think so. My concern was essentially, ``Is there any experimental (direct or indirect) evidence that gravity must be quantised in the first place?" The born-rule inconsistency seems to do the trick.

I am giving the SU(2) action. It acts trivially on X and in the usual manner on $\mathbb{P}^1$.

I agree with Scott Aaronson. In fact, the physicist way of defining tensors as things that change correctly under coordinates gives a nice way to define tensor fields on manifolds (Simply a smooth collection of multi-index beasts on different open sets such that on the intersection they are related by an appropriate transformation (the transition functions of the tensor bundle)). I am not sure if this "heuristic" actually gives rise to wrong intuitions.