For example, an accelerated electron will emit EM radiation, but will not emit gravitational radiation.
A common argument used is that the gravitational monopole represents the total mass-energy of a system. As this is conserved, there is no gravitational radiation. Why can’t we use the same argument to say that: the electric monopole represents the total charge of the electron, and as this is conserved, there should be no EM radiation?
In fact an accelerated mass always produces gravitational radiation. In the form of gravitational waves. This relativistic effect is usually negligible but it can clearly be noticed in binary black hole mergers since the masses involved are huge and the accelerations extreme. This gravitational radiation is what experiments like LIGO are able to detect since 2015.
Any mass, even the tiny amounts of mass electrons have, can produce gravitational waves if accelerated. I think you might be misunderstanding EM radiation as a way of a charge to get radiated away from an object, which doesn't happen, an EM wave is a perturbation on the strenght and/or orientation of the electric field.
Any system with accelerated quadrupole moment will emit gravitational waves.
Gravitational waves are emitted by all masses with accelerating gravitational quadrupole moments, but rarely with sufficient power to be detectable.
Do only black holes emit gravitational waves?
In reality, the second time derivative of the isolated system's stress-energy tensor must be non-zero to emit GWs.
You are saying talking about EM waves, and analogously, the changing dipole moment of a charge is neccessery for the emission of EM waves.
More technically, the second time derivative of the quadrupole moment (or the l-th time derivative of the l-th multipole moment) of an isolated system's stress–energy tensor must be non-zero in order for it to emit gravitational radiation. This is analogous to the changing dipole moment of charge or current that is necessary for the emission of electromagnetic radiation.
https://en.wikipedia.org/wiki/Gravitational_wave
Thus, an accelerated electron alone will not emit GWs. So you are correct, that the (non-uniformly) accelerated electron will emit EM waves, but no GWs.
Usually we need a system of masses to produce GWs.
You don't need a mass modulator, you just need something with a changing quadropole moment - the simplest example of this is a spinning dumbbell, and indeed this is basically what the binary pulsar system is.
Would it be possible to transmit information through gravitational waves?
Please note that according to the quadrupole formula, an (non-uniformly) accelerated electron that is emitting EM waves will not (necessarily) emit GWs.
I think what @Physics_noob is trying to say is that an electron, assuming it’s a perfect point charge, has zero quadrupole moment. This would indeed mean that there would be no GWs emitted.
At the moment, I have no comment on your second statement, where you talk about conservation of mass monopole vs electric monopole.
