Big problem with the standard derivation of Rayleigh-Jeans law + reason for this standard approach + solution

The problem:

The standard derivation of the Rayleigh-Jeans law, in a cubic reflective cavity with a small blackbody in it, would have you believe that the cavity has an energy density of $$kT$$ for every frequency that creates a perfect standing wave.

This doesn't make any sense since this would mean that the blackbody is radiating at a finite power per unit area for each of the specific frequencies associated with each standing wave:

(In case you are unfamiliar with the derivation: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/radpow.html#c1. This idea is actually pretty intuitive since absorptivity equals emissivity at thermal equilibrium.)

Since we can change the volume of the box this means that this is the case for every frequency...

So the blackbody is radiating a finite amount of power for every specific frequency and therefore radiates an infinite amount of energy for every frequency interval and will form a black hole as soon as it reaches a temperature of higher than absolute zero...

This is obviously nonsense.

It originates from the assumption that all the radiation in the box is contained in the perfect standing waves. I don't know why anyone would imagine that the blackbody will always be a good boy and only emit radiation that creates the perfect standing waves in the box or even know which frequencies these should be. In reality there is no mechanism that ensures that only and purely the perfect radiation persists in the cavity. Frequencies that are very close to the "perfect" ones will, as they form plane waves that bounce around the cavity to come back to their starting point, not overlap 100% perfectly with themselves but this does NOT mean that you can just wave your hand and POOF the associated energy density is zero! I do agree that frequencies that are too far from any "perfectly fitting" frequencies interfere so randomly and therefore destructively that they cannot build up and so can be ignored.

Why it is done this way:

Since the "almost perfect yet imperfect" frequencies don't form proper standing waves and are constantly changing as the EM radiation is bouncing around and interfering with itself in ever changing ways, you can wave bye bye to your breaking the field up in Fourier modes. This is a big loss because these modes were the perfect candidates for electromagnetic degrees of freedom of the system since there are countably many of them.

This forces us to make the additional assumption that the real degrees of freedom are the frequency bands around the perfect frequencies.

Perhaps this is not as straightforward given the definition of degrees of freedom as devised for a system of gas particles where things are much neater and clearer but this does not warrant all sorts of mental acrobatics and mathemagical gymnastics about why only perfect radiation should persists because it leads to big problems and is very confusing.