On the 1040STE the results were (only) partly promising. The lines were visibly reduced, but contrary to my MegaSTE they didn't fully vanish. As requested on FB, I want to document my findings, though, while I think of solutions for the 1040STE.
After a lot of measuring, I've come to the conclusion that the lines you see are caused by impedance coupling from the power supply on the mainboard. As you might know, the STE/MegaSTE Shifter contains the multiplexers/latches to moderate between memory access for the CPU and for video output. When these multiplexers switch, momentarily an increased current flows through Vcc and GND, i.e. the IC's power supply pins. This causes a phenomenon known in literature as ground (or Vcc) bounce
where the supply rails within
an IC change in voltage.
Obviously, such a change of the IC-internal supply rails will be overlaid on any output signal (e.g. see Figure 2 in the PDF linked above). This is far less critical on purely digital outputs (as long as no high/low threshold is crossed). But the video output signals from the Shifter (including the high-res one) are essentially treated as analog signals and so any ripple on them will be visible. Because the switching of the multiplexer occurs every 500 ns (1/2 MHz) you can see the effect every 16 pixels in high-res mode.
Since this is an effect internal to the Shifter, no amount of recapping will fix it -- different from other power supply related effects. Hence, my fix consists of a digital buffer that will take the mono video signal as input and output a ripple free version of it -- assuming the buffer itself is supplied from a clean power supply, such as directly from the big buffer capacitor on the mainboard.
As this fixes the issue on the MegaSTE, I'm confident I'm on the right track. Currently, I think that in the 1040STE also power supply variations outside of the Shifter (e.g. at the transistor which the high-res signal passes) overlay further ripple and thus also cause (less visible) vertical lines.