Why do accelerating anodes work?
After leaving the electron gun, electrons are in an electric-field-free space...in every direction they see high voltage equal to the voltage of the electron gun's final anode. So they travel in a straight line, constant velocity.
Until they hit the phosphor screen.

After that, they are collected by the conducting aquadag coating, which is also at the same high-voltage of the final anode. From there, they return (as beam current) to the high-voltage DC supply. Some glass may have no aquadag - then electrons find their way to the final anode.

A test setup had one of these old tubes run with a variable +high-voltage anode supply. At low accelerating voltage, electrons sometimes would "stick" to the phosphor when they hit. Perhaps a section of phosphor that was electrically isolated - it is a thin coating. That section became charged, and the electron beam would not go there - it became dark and wouldn't glow.
 If anode voltage was raised, the higher-speed electrons would un-charge that spot, and instantly, it would light up as it should. This dark-region-thing would only happen when the accelerating voltage was much lower than the intended operating voltage.