Szczepan Bialek wrote:
Also, if the earth is absorbing your electrons, something should be
gaining a rather large positive charge as a result of the
transmission. Where is the positive charge?

Without the "infinite source or sink for charge" a transmitter is gaining a
rather large positive charge.

It is a pity that you are not a radio amateur yourself.
If so, you could see with your own eyes (or feel with your own hands)
that this is not true, and so the whole theory is incorrect.

But maybe you have a cellphone. Try making a long call while you stand
on one of those glass tables that they use in static electricity
experiments.
See if you develop a positive charge while making that call.
(and not doing other things like stroking your cat)