I’ve given more thoughts about this observation. My current conclusion is that the claim that vanishing Neumann boundary conditions are applied by default if no boundary conditions are explicitly specified, for a given finite element function, is not correct. And the above MWE is indeed a case where this happens.
If you go through the hand derivation of the weak form of the PDE \nabla \cdot \vec J (equivalently, if you integrate this equation over the whole volume and then apply Gauss theorem), you’ll find a different expression than the one I posted in my MWE. In fact, in my MWE, it is exactly the same problem except that I enforce a given electrical current. By doing so, \nabla V has to adjust such that the input and output currents match what I prescribe on the boundaries. And \nabla V does not vanish on all boundaries, in particular those where Dirichlet boundary conditions are specified for the other unknown (temperature in this case).
This also responds to Implementing a potential problem with a current condition - #21 by Daniel.Baker. I do not believe prescribing the current on the 2 boundaries (in and out current) is insufficient to fully specify the voltage (scalar field). I believe it is perfectly fine, keeping in mind that voltage (or electrostatics potential) is defined up to a constant.
I would appreciate if @kamensky could comment.