I am entering this post in hopes of getting some additional posts from our other members that might shed some light on this question.
I know you looking for a resolution of the method used by AWS and I can't give that to you; but, with regard to your post I have a couple of questions/comments.
In your description of the text book solution you stated that the "normal force" was divided by 2. I assume you meant the "normal stress". Is that correct?
With regard to the dividing of the normal force/stress by 2, that is not something I can understand. Beyond that the sqrt of the sum of the squares part is simply resolving the perpendicular normal stress and shear stress vectors into their single combined vector.
As to the conflict of the normal stress vs. shear stress terminology between the text and AWS; I think that they may both referring to the same thing but since, as you stated, "for fillet welds all stresses are considered shear stresses" it may be that AWS simply decided to utilize that term for the normal stress on the weld. I can somewhat understand that because if you think about the angular orientation of the weld throat then the moment loading clearly creates a shearing force on that x-section; and the same situation applies to the direct shearing load as well.
As for the combining of the shearing load and tensile load stresses, one thing I noticed in my Machinery Handbook and other texts is for round cantilever beams all display the moment stress solutions but none of them address the point loading; and, during rather rigorous Googling I still could not I find any academic or other sources, that address all types of cantilever beam and welded beam issues, that the address point loading condition.