Pokemon on the German ZX Forum has designed a number of video improvement circuits (ZX81CPP, ZX81SCP) that allow connection of a composite video monitor and produce a crystal clear picture once properly adjusted.
https://www.sinclairzxworld.com/viewtopic.php?f=8&t=841&sid=9e23e93b43bfcc0bb782a80803430e70
The circuit basically completely re-creates the video signal from the ULA output.
Newer versions have an ATTiny(I think) on board that keeps the sync alive during fast mode which is a relief on modern LCD screens as they tend to get upset otherwise.
Every once in a while he offers PCBs and completely built circuits (which, BTW, nicely fit into the modulator case if cleaned up a bit).