Does classical electromagnetism really predict the instability of atoms?

The essential conceptual problem in your treatment is the fact that you assume that the radiation should somehow emerge solely from particle $A$ acting on particle $B$. This is not true, the radiation (and radiation-reaction) comes from the particle $B$ acting on itself! To understand this, you must consider $A$ and $B$ to be finite bodies first. Then the radiation schematically emerges as:

  1. all the charges in the body $B$ "sending out" their own electromagnetic potential on their null cones $|\vec{r}-\vec{r}_B|-ct = 0$,

  2. body $A$ accelerating charges within body $B$,

  3. and finally, the accelerated charges within $B$ interacting with some of the electromagnetic potential from within $B$ (point 1.)!

(You can swap $B$ and $A$ to get the radiation from particle $A$ as well.)

When the dust settles, you can take a limit of the sizes of the bodies going to zero and you get the famous Abraham-Lorentz-Dirac force acting on either $A$ or $B$: $$F^{\mathrm{ALD}}_\mu = \frac{\mu_o q^2}{6 \pi m c} \left[\frac{d^2 p_\mu}{d \tau^2}-\frac{p_\mu}{m^2 c^2} \left(\frac{d p_\nu}{d \tau}\frac{d p^\nu}{d \tau}\right) \right]$$

Nevertheless, the derivation of this result is notoriously challenging both conceptually and technically. The reason for that is that when you treat the body $B$ as an infinitely small particle, it should not be able to interact with data on its own light-cone because that would mean it is moving beyond the speed of light! On the other hand, the particle is on its own lightcone at $t=0$ and $\vec{r} = \vec{r}_B$, and the potential and the Lorentz force diverge at that exact position!

The only way to resolve this in a rigorous way is to assume, as already mentioned above, that the bodies in questions are of finite spatial extent and finite charge densities. Then, you take the limit of the size going to zero. This was carefully redone in 2009 by Gralla, Harte & Wald and I recommend that paper for further information. (The reason why this topic has received heightened interest recently is the fact that gravitational-wave inspirals of small stellar-mass astrophysical objects into supermassive black holes can be treated exactly in the approximation of a "self-forced particle", see Barack & Pound, 2018.)

You can derive the Larmor formula from a particular perturbative approximation of the ALD force called order reduction. First you take the $\mathrm{d}p^\mu/\mathrm{d}\tau$ for the particle without radiation reaction and insert it into $F^{\rm ALD}_\mu$. Then the Larmor formula is just the rate with which this force is taking energy from the particle.

EDIT: A broader historical discussion

Jan Lálinský reminded me that there exist formulations of classical electrodynamics that 1) agree with most of the predictions of Maxwell equations in the continuum limit (given a certain "absorbing universe" postulate), and 2) where the "point particle" is not the limit of a finite body and does not feel any self-force. A brief review of these "Schwarzschild-Tetrode-Fokker(-Frenkel)" (STF) electrodynamics was given by Wheeler & Feynman in 1949.

Depending on how exactly do you implement the "absorber universe", the quasi-neutral ensemble of particles far away from your system, the planetary atom is also usually unstable in STF electrodynamics. This is because the energy tends to be stolen from the atom by the ensemble of far-away particles and dissipated (even though possibly at a slower rate). On one hand, this is a nice "Machian" twist on electrodynamics, since the notion of the field is emergent from the physical particles, and the particles would not radiate energy if there were no other particles to pass the energy to. On the other hand, the STF electrodynamics tend to have curious non-local properties such as the absorber universe "knowing" about an action on the particle an infinite time before the action itself occurs! This makes the theory physically unsatisfactory to me.

Consider the following example of a pulsar, whose pulse we detect and whose rotation rate slows down as a consequence of the radiative energy loss. In mainstream electrodynamics, we talk about electromagnetic waves traveling for eons through space from the pulsar as independent energy-carrying entities, while the STF theory defies this picture. While in the mainstream electrodynamics the wave took the energy away from the pulsar and made it slow down its rotation rate, in the STF theory the pulsar slows down (or not) thanks to the fact that it "knows" that energy-receiving objects such as your antenna will be there in a thousand years!!!

Ultimately, both the usual particle+field and STF theories are wrong, and the correct theory of electrodynamics of fundamental point particles is quantum electrodynamics (and even more ultimately the Standard model), so this is more of an academic discussion. However, I find the STF picture grossly undidactic as compared to the understanding of classical electrodynamics as the theory of the electromagnetic field sourced by finite continua that we sometimes limit towards approximate point particles.

John Lighton Synge had similar idea and he analyzed numerically the equations of motions for two oppositely charged particles of arbitrary masses where only retarded EM forces are present.

J. L. Synge, On the electromagnetic two–body problem., Proc. Roy. Soc. A 177 118–39 (1940)

He found that the system still collapses, but the greater the difference in masses, the more slowly it does so. For hydrogen atom, the time of collapse turned out to be hundreds of times longer than the time naively obtained from the Larmor formula.

The collapse is due to simple but perhaps too simple assumption: that the force acting on any particle is just retarded EM force due to the other particle. Then, because the motion of the particles is anticorrelated (when one moves left, the other moves right), the system radiates EM energy away.

If we introduce into the model background EM radiation that acts on both particles (additional forces), collapse ceases to be inevitable, because the radiated energy can be supplied by the background radiation forces. There are some papers on that - for more on this, see also my answer here:

The classical electrodynamic atom