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As pointed out by Dr jh in his comment, the planetary model of the atom is wrong. Classical electrodynamics predicts that if electrons really did orbit the nucleus like little planets, the orbital motion would cause the electron to radiate its orbital kinetic energy away as electromagnetic waves and then fall all the way down into the nucleus- just as you say. Since this does not happen in reality, the model is incorrect.
There is no way to save this model "classically" because there is no classical mechanism by which to halt the orbital collapse of the electron. Quantum dynamics does, by establishing a ground state orbital energy level with no energy levels below it into which the electron could transition. Thus far it falls, and no farther.
Lord Raleigh determined the size of carbon atoms 1890-1900 by measuring the spread of a single layer of oleic acid on water, allowing estimating the size of an atom. By 1911 Rutherford had determined that the nucleus was about 10,000 times smaller than the atom as a whole. He proposed the planetary model about this time.
Arguing that the electron would spiral in and come to rest against the nucleus clearly doesn't work because of the size discrepancy. If one were to postulate that electrons are much bigger than nuclei the cross-sections and deflections found in Rutherford's experiment would not make sense.
The standard textbook explanation for the instability is just that in electrodynamics a charged particle moving fast in a circular orbit will radiate away energy. One could try to save things by suggesting that Maxwell's equations do not apply on the atomic scale. I have not seen any example of this being seriously proposed, but it is certainly a possibility - but an ugly one, especially to 1910s classical physics (especially since it would cast doubts on interpretations of the observations, which were implicitly Maxwellian). Still, Weber had an earlier and not very well-known theory of charges forming "molecular" atoms that involved a slightly altered electrodynamics.
The Bohr model explained discrete emission lines but still did not explain why there was no inspiral; this probably helped people make the jump to a quantized view where only some photons could be emitted.
One can apparently construct theories where the electron is a classical field instead of a particle, (Rashkovskiy 2016) gives an example. This is decidedly non-mainstream today and actually requires using the Dirac equation that came from quantum theory, but I can imagine some alternate history where early 20th century physics tried to patch the planetary model by electron-field waves - except that it looks to me that it would also quickly lead to the jump to the quantized view.
In short, one can always propose solutions to the stability problem, but solutions also need to make sense with the rest of physics. That makes many classical stability solutions look very awkward.
