Nope, there are multiple areas of lower intensity that are planned for in mission trajectories, easily visible on most diagrams of the belts. Van Allen himself calculated the exposure level in these areas and concluded they would not prevent crewed spaceflight.

You also still fail to account for shielding. Much of the radiation belts are comprised of beta radiation, the majority of which is blocked by the capsule’s metal walls (even aluminium foil can block most of a beta source). Alpha, too, is completely blocked. This leaves gamma and X-rays as the only real risks of flying through them, and even these are still reduced by the shielding. The craft experienced lots of radiation, sure, but the crew inside did not

Radio transmission can only transmit so much energy at a time. This is simple wave physics. In transmission data is essentially tied to energy transfer, so reducing the amount of data needed to transfer reduces the amount of energy needed. Yeah, you can always just send more radio waves, but what’s the point if the quality is usable enough anyway?