The answer I think is that you have to compare like with like.
Let's take field propagating from the accelerated charge at the origin. Once the field is in the vacuum, the Maxwell's equations are time-reversible, so you could simply 'play the video backwards' and see the waves converging into a point. BUT, once the waves get to the origin, in your video playback, there must be a charge there. If there is one, good, the charge will simply absorb the power (where would the power go? no-where. you did not specify how you made the charge move in the first place - that is the fudge-factor).
What if there is no-charge at the origin? What if you built a spherical distribution of antennas aiming to launch the field back into the origin with the same distribution as that from an accelerated charge, but there is no charge there? My intuition tells me that this would not be possible, i.e. you would find it impossible to launch just that field and nothing else. Because that field would not be a solution to Maxwell's equations.
To go any further, I think you need to specify more precisely what field would you launch.