The ‘milky’ oil IS caused by short runs, a water cooled engine requires at least 8 miles to warm up fully, only then will the condensation burn off.
Doesn’t the clutch cable on a ZZR go to the clutch side, not into the drive side of the engine? (could be wrong here, it’s been a while) if it does then you have no pushrod, so no adjustment there. If the clutch arm is on the clutch cover then check to see what angle it’s at, it wants setting so it’s just slightly greather than 90 degrees with the clutch cable holding boss, this will give you the greatest leverage. Also where is the span adjuster set on your clutch lever? Set it so the lever is furthest from the bars, it isn’t meant to alter the action of the clutch but it does.
A drive chain that’s too tight will make the problem feel worse too.
Don’t put thinner oil in, it really won’t help your cause in the long run.