Some people object the idea of reciprocating ExH waves in a charged capacitor because they claim that then we should have ohmic losses. Why?

Why should we associate the emergence of ohmic losses with the basic ExH energy current propagation. At this level there is no point to talk about ohmic losses because at this level we have no idea about any charge movement, i.e. no electric current is defined here. It’s all about EM energy current.
Ohmic losses need only be considered at the level of the superposition of Electric and Magnetic fields, i.e. at the level of V and I values in a particular place in space and a particular point in time.
If we take any of Wakefield experiments, we can find in them an interval of time at a certain place where the odds of overlap of two reciprocating waves are such that they produce no current (i.e. the cumulative effect on magnetic field is zero), so there are ohmic losses here and then. This does not however preclude the two ExH waves to move in opposite directions.