How to always have the local Z axis pointing upwards in the viewport? I am editing an object, use local transform orientation.

Here it is



Why is not Z, the blue arrow pointing "screen upwards"?

That would make life so my easier!
World Transform and Local Transform exists for reasons.

You can reset/apply your object's transform to make your Local Z axis pointing upward.  But there is no way to lock Z axis to point upward.  What you can try is to move/rotate your object's mesh in Edit mode, in there, your rotation/move operations won't change the local axis's orientation.