Could be wear on the bearing housing. This can happen when the bearing ball seizes on the rudder shaft and starts turning in the housing instead. Or could be the upper housing never was the right size, or the new bearing the wrong size for it, however you want to put it.
In my case the original rudder shaft wasn't manufactured according to plans, so ended up sleeving it to fit the new Jefa bearing (that was made to plans). In your case the play is between the bearing ball and the housing, which isn't so easily fixable due to the spherical surface.
To fix, you need either a bearing ball that perfectly fits the housing you have, or a new housing that fits the new Jefa (presumably roller) bearing.
You're right that the shaft isn't a tapered design (it's stepped instead), so tightening the nut only pulls the rudder up but doesn't reduce play anywhere (unless you crank it so far up you hit one of the steps, which won't help for long).