But doing so would just converge dE even quicker, still leaving dH trailing, no? Because "per atom" would mean dE/N but still H. Or did I miss your point?
I think it's more a question that the band structure energy (note that E is not the DFT total energy, it's the weighted sum of occupied states) becomes stable but the SCF loop still needs to move some charge between different orbitals to get a converged density matrix/Hamiltonian. So you could have an "almost-converged" state where the total energy is ok, but things like forces and even band structures may not look so good, if you interrupt earlier. One should also note that the default convergence criterion 4e-5 in ATK is actually quite lax, for really good forces etc one should typically go down an order of magnitude more.