Hello Robert, I have often played with the mesh warp tool, but was never really concerned about 'precise' angles. I just grabbed the text after using the warp tool and rotated to 'roughly' the angle I wanted. So, your method using a straight line and then lining-up might be the only way for precise angles? But, looking at all the warp shapes, there are not many that might fit your bill, save the top one I have highlighted and its neighbour to the right that might need precise angles - as most of the others are intricate shapes, as the example I highlighted below them. Although having said that, any of the shapes with a straight line component in its shape would be suitable for angle precision.
In any event I did have a play in X9 with the top highlighted warp shape and when you click the shape in rotates at pretty much 4.5 dgrees up to the right - I then produced the straight line (your method) rotated the line to 8 degrees (which is the angle I wanted) and then set the warped text to match at 8 degrees. I can't see any other way to get precise angles, other than your method.