We have to clear up the concept of "body radiates/body does not radiate".
Accelerated charged bodies do radiate, or "have electromagnetic field that had radiation-like characteristics" but not in a way a hot body does.
Hot body sends out chaotic electromagnetic field whose intensity falls off as $1/r$ in all directions, with random spatial fluctuations on the scale of IR wavelengths. Due to randomness of this EM field, it is detectable by observers at rest in all frames of reference, the radiation cannot be "transformed away everywhere" for an observer having appropriate acceleration (except when the observer is far enough from the body, then the radiating body can be behind the Rindler horizon, but let's ignore that for now).
Uniformly accelerated charge has very orderly EM field that is very different from EM field of a hot body. It does not have fluctuations, oscillations, it changes only slowly on the time scale it takes for the body to approach and leave the region of the observer. Hypothetically, such smooth orderly field can be transformed away by change of reference frame. This is what most probably happens when observer gets in the same reference frame as the accelerated charge. So yes accelerated charge "radiates" but only in the frame where it is accelerated.
Now let's look at the last example, a charged body stationary with respect to Earth's ground. We do not expect this body to have radiation field in Earth's frame, for the simple reason, the charged body has zero acceleration in this frame. Radiation field could be present in other frames, for example in the frame of free-falling observer that passes close to the charged body.
Suppose that is so. Where does the energy of radiation come from?
Well, consider energy of any body in this frame - kinetic energy of the charged body, kinetic energy of the Earth. All are increasing in time as the observer falls. This is immense amount of energy being gained in time from no apparent source.
This violation of conservation of energy is traditionally understood to be due to observing things from the "wrong frame" of accelerated observer.
The problem does not go away if the acceleration is due to gravity, like for free-falling observer.
In such frames we can introduce fictitious forces acting on the Earth body and the charged body. But there is no apparent source of these fictitious forces and no apparent source of corresponding energy gained by those bodies. So the problem with energy conservation in such frames is already with kinetic energies of solid bodies and already in non-relativistic theory.
In relativistic theory including gravitation this problem is even worse. There can be things such as bodies accelerating and EM energy increasing without apparent source of energy. There are also things such as Rindler/black hole horizons where things disappear or freeze in time. There are difficult problems and energy conservation is one of them.