why is this Hamiltonian taken to explain the lack of radiation friction in QM, whereas in classical electrodynamics it is an invalid Hamiltonian for that exact same reason?
That Hamiltonian is not an explanation of "lack of radiation friction in QM". It is an Hamiltonian that does not manifest that friction, neither in classical nor in the quantum theory, because it is based on a fiction - an instantaneous Coulombic interaction, where no EM radiation exists.
The reasons textbooks suggest Schroedinger's model resolved the problem with stability of the atom are not entirely clear to me, but it is probably partially because:
Schroedinger's model proved to be very general and successful, not only for atoms, but molecules;
Schroedinger's model has a ground state, which the older classical model based on EM theory has not.
With these observations, it is natural to expect that whatever happens with EM interaction in the atom, the atom cannot collapse in quantum theory, because the immensely successful Schroedinger's model says its energy cannot go below certain value.
Of course, careful student will notice that this argument is unsatisfactory, because the Hamiltonian used is simplistic. It does not even obey special relativity, far from taking into account fine details of EM interaction such as EM radiation.
To answer questions related to stability of the atom, both in classical and quantum theory, in a satisfactory way, one has to include the revelations of special relativity, such as the fact that the interaction cannot be instantaneous. The most plausible direction then is to assume Maxwell's equations hold down to the atomic level and go from there, but the calculations aren't easy.
The question of stability also needs to be stated in a more specific way, including some specification of the environment the system is in, for example by stating the state of the external EM field, and how exactly are the system's particles' fields connected to their motion - whether they are retarded, advanced, or some mix of the two plus some free field component. What the old and common accounts of this problem forget (including Bohr's) is that real atoms are not it empty featureless vacuum, but they are under constant action of background EM fields, from other atoms nearby and EM radiation coming in from far distances. A molecule of hydrogen in an empty universe filled with thermal radiation may not be stable (because the radiation will break it down and the parts will go their separate ways), but in a chamber full of high pressure hydrogen it may be (because the parts cannot easily go their separate ways, due to presence of other molecules).