They are being sent, yes; but the gain is so low that nearly none of them are hitting often enough to light up the phosphors.

It's the phosphors on the back of the front-glass that matter. They are what "burn in". The dimmer knob actually turns the gain down on the electron gun, so that it sends out fewer. A dimmer screen is less prone to burn-in. (And even if it does, it burns in with less intensity; which can be overcome by displaying an opposite image at higher brightness for a shorter period of time. Of course, then you're just burning in the rest of the screen to match; which results in a dimmer screen overall.)