Most tensor subspaces of low dimension have rank-1 defining equations

Let $$V_1,\ldots , V_k$$ be vector spaces of dimensions $$n_1,\ldots , n_k$$ over a field of characteristic zero.


(See eg. Prop 1.3.3 in Russo's "Geometry of Special Varieties." The statement assumes characteristic $$0,$$ though I'm not sure if it's necessary for me.)

Thus, if $$\Phi (\bs{\alpha} ) = \Phi (\bs{\alpha}')$$ for $$\bs{\alpha}$$ generic, then $$\bs{\alpha }$$ equals $$\bs{\alpha}'$$ after permuting some of the $$j$$ factors in the domain.

I discovered the above argument for the case where $$k=2$$ and $$(n_1, n_2) = (3,3),$$ in a context apparently unrelated to tensors, and was surprised that it generalized with little effort. I am thus wondering if it is a well-known fact or if there is some simpler explanation.

A seemingly more difficult question I am also interested in: describe the constructible set $$\im \Phi .$$