In this work, we establish a link between the small-time approximate controllability of bilinear Schr\"odinger PDEs (posed on a boundaryless Riemannian manifold $M$) and the control in the group ${\rm Diff}_c^0(M)$ of the diffeomorphisms, isotopic to the identity and with compact support, of the underlying manifold $M$. More precisely, under a density assumption on the Lie algebra generated by the control potential and the Laplacian, we show that compositions $|J_P|^{1/2}(\psi_0\circ P)$ of the initial wavefunction $\psi_0\in L^2(M,\mathbb{C})$ with any diffeomorphism $P\in{\rm Diff}_c^0(M)$ can be approximately reached, in arbitrarily small times, by controlled solutions of the Schr\"odinger equation (here, $|J_P|$ denotes the determinant of the Jacobian of $P$). We illustrate this property on two examples, posed respectively on the torus $\mathbb{T}^d$ and on the euclidean space $\mathbb{R}^d$. As a physical application, we obtain in particular the small-time approximate control of the quantum particle's averaged positions. This yields also new small-time approximate controllability properties between families of eigenstates on $\mathbb{T}^d$. To prove the result, we first construct solutions of the Schr\"odinger equation that approximately evolve, arbitrarily fast, along any unitary transport flow on $L^2(M,\mathbb{C})$. In this way, we control the composition with any diffeomorphism that can be decomposed as a product of flows on $M$. We then combine this property with a result of Thurston on the simplicity of the group ${\rm Diff}_c^0(M)$ to conclude.