More info here, skip down to Section 5, and following:
http://www.oestex.com/tubes/screens.htmWhat I'm getting is this: Quoting the old tube handbooks (which did not envision signal distortion as a desirable feature), a regulated screen supply is recommended, especially for Class AB operation. The reason is that the screen grid, properly charged with a B+ voltage, makes fluctuations in
plate voltage almost immaterial, so the tube stays linear. But fluctuations in
screen voltage hugely affect tube current and linearity. If the screen voltage is held constant, then a smaller input signal voltage produces more gain with more clean headroom and pleasing harmonics.
The zener diode is a simple voltage regulator. For the above BJr. screen circuit mod: If if the zeners drop and regulate the B+ supply to the screens enough, they might be able to more or less regulate the screen supply over the range of fluctuation of B+ voltage in the amp. According to the article I quoted, that should yield more clean headroom with pleasing harmonics.
EDIT: Alternatively, the plate & screen supplies could be well-regulated. But the simple zener mod might yield some of these results.This would be the exact opposite of the KOC mod which calls for large screen resistors instead of voltage regulation.